glen/loops
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

update to LOOPS 3.4.0

Browse Source 
These are simply the changes as distributed.
This commit is contained in:
Glen Whitney 2017-10-29 23:54:13 -04:00
parent 7e8b3b5562
commit f64208f12f
58 changed files with 17724 additions and 29097 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

8
PackageInfo.g
View File

@ -1,9 +1,9 @@
SetPackageInfo( rec(
PackageName := "loops",
Subtitle := "Computing with quasigroups and loops in GAP",
Version := "3.3.0",
Date := "26/10/2016",
ArchiveURL := "http://www.math.du.edu/loops/loops-3.3.0",
Version := "3.4.0",
Date := "27/10/2017",
ArchiveURL := "http://www.math.du.edu/loops/loops-3.4.0",
ArchiveFormats := "-win.zip .tar.gz",
Persons := [
@ -83,7 +83,7 @@ Dependencies := rec(
),
AvailabilityTest := ReturnTrue,
BannerString := "This version of LOOPS is ready for GAP 4.7.\n",
BannerString := "This version of LOOPS is ready for GAP 4.8.\n",
Autoload := false, # false for deposited packages
TestFile := "tst/testall.g",

41
data/automorphic.tbl
View File

@ -2,39 +2,21 @@
##
#W automorphic.tbl Automorphic loops G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: automorphic.tbl, v 3.3.0 2016/10/20 gap Exp $
#H @(#)$Id: automorphic.tbl, v 3.4.0 2017/10/23 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
## Binding global variables
## LOOPS_automorphic_cocycles
## LOOPS_automorphic_bases
## LOOPS_automorphic_coordinates
# Many small automorphic loops are represtented by encoded Cayley tables.
#
# Commutative automorphic loops of order 243 are represtented as central
# extensions of the cyclic group of order 3.
# The necessary data is only loaded on demand and consists of:
# - LOOPS_automorphic_cocycles, a list of encoded bases of the
# space of cocycles modulo coboundaries for every factor loop F needed.
# - LOOPS_automorphic_coordinates, a list that for every loop
# points to the factor loop and gives coordinates of the required cocycle
# with respect to the relevant basis.
LOOPS_automorphic_data := [
#implemented orders
[3,6,8,9,10,12,14,15,27,81,243],
[3,6,8,9,10,12,14,15,27,81],
#number of nonassociative loops of given order
[1,1,7,2,3,2,5,2,7,72,118451],
[1,1,7,2,3,2,5,2,7,72],
#the loops
[
#order 3 (Z_3)
#order 3 (Z_3, use left Bruck loops, placeholder only)
[
"201"
],
#order 6
[
@ -50,10 +32,8 @@ LOOPS_automorphic_data := [
"0325476301675421076455760132467102374523106543201",
"0325476310674520176545761023467013275432106452301"
],
#order 9 (two abelian groups)
#order 9 (two abelian groups, use left Bruck loops, placeholder only)
[
"204537861534867678012861207201345534",
"204537861534867678120862017012453345"
]
,
#order 10
@ -80,20 +60,13 @@ LOOPS_automorphic_data := [
"234068597BDAEC340189675DEBCA401297856ECDAB012375968CAEBD6897ADECB041328975DCABE430215689EABDC102439756CBDEA324107568BECAD21304BDEC0413258976DECA4302187569ABDE1024395687ECAB3241076895CABD2130469758",
"234067895BCDEA340178956CDEAB401289567DEABC012395678EABCD7968ADBEC012348579ECADB340129685DBECA123405796CADBE401236857BECAD23401DBEC0432156789ECAD3210478956ADBE1043295678BECA4321067895CADB2104389567"
],
#order 27 (commutative only, placeholder)
#order 27 (commutative only, use left Bruck loops, placeholder)
[
]
,
#order 81 (commutative only, placeholder)
[
]
,
#order 243 (commutative only, placeholder)
#order 81 (commutative only, use left Bruck loops, placeholder)
[
]
]
];
LOOPS_automorphic_cocycles := [];
LOOPS_automorphic_bases := [];
LOOPS_automorphic_coordinates := [];

8058
data/automorphic/automorphic_cocycles.tbl
View File

File diff suppressed because it is too large Load Diff

61
data/cc.tbl
View File

@ -1,31 +1,52 @@
#############################################################################
##
#W cc.tbl CC-loops p^2, 2p, for p odd prime G. P. Nagy / P. Vojtechovsky
#W cc.tbl Library of CC loops G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: cc.tbl, v 3.0.0 2015/06/10 gap Exp $
#H @(#)$Id: cc.tbl, v 3.4.0 2015/06/10 gap Exp $
##
#Y Copyright (C) 2005, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
## Binding global variables
## LOOPS_cc_used_factors
## LOOPS_cc_cocycles
## LOOPS_cc_bases
## LOOPS_cc_coordinates
# CC loops are activated as follows:
# If n = 2p or p^2, where p is a prime, then we call a method for
# cosntructing these loops.
# If n = 2p, where p is an odd prime, then we call an algebraic method for
# constructing these loops.
# If n = p^2, where p>3 is a prime, then we call an algebraic method for
# construction these loops.
# If n is a power of 2 or 3, then we use cocycles located in cc/cc_cocycles_n.tbl.
# For all other orders, we point to the library of RCC loops.
LOOPS_cc_data := [
#implemented orders
[ 8, 12, 16, 18, 20, 21, 24, 27],
#number of nonassociative loops of given order
[ 2, 3, 28, 7, 3, 1, 14, 55],
#the numbers of the loops in the RCC library
[ 2, 3, 4, 5, 7, 8, 9, 12, 16, 18, 20, 21, 24, 25, 27, 32, 49, 64, 81, 125, 343],
#number of loops of given order in the library
[ 1, 1, 2, 1, 1, 7, 5, 3, 42, 7, 3, 1, 14, 5, 60, 437, 5, 14854, 5406, 84, 122],
[
#order 8
[2,7],
#order 2 (Z_2)
["010"],
#order 3 (Z_3)
["201"],
#order 4 (placeholder only)
,
# order 5 (Z_5)
["2340401123"],
# order 7 (Z_7)
["234560456016012123345"],
#order 8 (placeholder only)
,
#order 9 (placeholder only)
,
#order 12
[53,73,89],
#order 16
[9,35,107,228,243,292,437,440,1043,1883,1936,2332,2420,2636,2645,2750,2753,2794,2797,2847,3682,3730,3739,3848,3949,4735,4904,4925],
#order 16 (placeholder only)
,
#order 18
[22,29,77,292,360,377,1133],
#order 20
@ -33,18 +54,12 @@ LOOPS_cc_data := [
#order 21
[104],
#order 24
[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216],
#order 27
[78,86,317,319,361,571,711,1080,1085,1624,1665,2217,2219,3614,3624,8579,8582,15059,15072,15503,15512,19439,23177,23214,26331,26348,52978,55027,55055,59116,59123,75864,78970,79011,83042,83104,83155,104913,106081,106144,110854,110892,110930,114102,117212,119407,134858,136370,140791,148160,148892,149330,151792,152090,152515]
[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216]
]
];
# The following can be used to point to CC loops of order 2p and p^2 in the library of RCC loops.
# order 6, [3]
# order 9, [5,4,3]
# order 10, [16]
# order 14, [97]
# order 22, [10346]
# order 25, [86,93,118]
# order 26, [151964]
LOOPS_cc_used_factors := [];
LOOPS_cc_cocycles := [];
LOOPS_cc_bases := [];
LOOPS_cc_coordinates := [];

61
doc/chap0.txt
View File

@ -6,7 +6,7 @@
Computing with quasigroups and loops in GAP
Version 3.3.0
Version 3.4.0
Gábor P. Nagy
@ -28,7 +28,7 @@
-------------------------------------------------------
Copyright
© 2016 Gábor P. Nagy and Petr Vojtěchovský.
© 2017 Gábor P. Nagy and Petr Vojtěchovský.
-------------------------------------------------------
@ -167,10 +167,12 @@
6.11-3 QuasigroupsUpToIsomorphism
6.11-4 LoopsUpToIsomorphism
6.11-5 AutomorphismGroup
6.11-6 IsomorphicCopyByPerm
6.11-7 IsomorphicCopyByNormalSubloop
6.11-8 Discriminator
6.11-9 AreEqualDiscriminators
6.11-6 QuasigroupIsomorph
6.11-7 LoopIsomorph
6.11-8 IsomorphicCopyByPerm
6.11-9 IsomorphicCopyByNormalSubloop
6.11-10 Discriminator
6.11-11 AreEqualDiscriminators
6.12 Isotopisms
6.12-1 IsotopismLoops
6.12-2 LoopsUpToIsotopism
@ -256,28 +258,31 @@
9.2 Left Bol Loops and Right Bol Loops
9.2-1 LeftBolLoop
9.2-2 RightBolLoop
9.3 Moufang Loops
9.3-1 MoufangLoop
9.4 Code Loops
9.4-1 CodeLoop
9.5 Steiner Loops
9.5-1 SteinerLoop
9.6 Conjugacy Closed Loops
9.6-1 RCCLoop and RightConjugacyClosedLoop
9.6-2 LCCLoop and LeftConjugacyClosedLoop
9.6-3 CCLoop and ConjugacyClosedLoop
9.7 Small Loops
9.7-1 SmallLoop
9.8 Paige Loops
9.8-1 PaigeLoop
9.9 Nilpotent Loops
9.9-1 NilpotentLoop
9.10 Automorphic Loops
9.10-1 AutomorphicLoop
9.11 Interesting Loops
9.11-1 InterestingLoop
9.12 Libraries of Loops Up To Isotopism
9.12-1 ItpSmallLoop
9.3 Left Bruck Loops and Right Bruck Loops
9.3-1 LeftBruckLoop
9.3-2 RightBruckLoop
9.4 Moufang Loops
9.4-1 MoufangLoop
9.5 Code Loops
9.5-1 CodeLoop
9.6 Steiner Loops
9.6-1 SteinerLoop
9.7 Conjugacy Closed Loops
9.7-1 RCCLoop and RightConjugacyClosedLoop
9.7-2 LCCLoop and LeftConjugacyClosedLoop
9.7-3 CCLoop and ConjugacyClosedLoop
9.8 Small Loops
9.8-1 SmallLoop
9.9 Paige Loops
9.9-1 PaigeLoop
9.10 Nilpotent Loops
9.10-1 NilpotentLoop
9.11 Automorphic Loops
9.11-1 AutomorphicLoop
9.12 Interesting Loops
9.12-1 InterestingLoop
9.13 Libraries of Loops Up To Isotopism
9.13-1 ItpSmallLoop
A Files
B Filters

64
doc/chap0_mj.html
View File

@ -31,7 +31,7 @@
<h2>Computing with quasigroups and loops in <strong class="pkg">GAP</strong></h2>
<p>Version 3.3.0</p>
<p>Version 3.4.0</p>
</div>
<p><b>Gábor P. Nagy
@ -50,7 +50,7 @@
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2016 Gábor P. Nagy and Petr Vojtěchovský.</p>
<p>© 2017 Gábor P. Nagy and Petr Vojtěchovský.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
@ -331,10 +331,12 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
</span>
@ -474,60 +476,66 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8028D69A86B15897">9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.4 <span class="Heading">Moufang Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.5 <span class="Heading">Steiner Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.5 <span class="Heading">Code Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.6 <span class="Heading">Conjugacy Closed Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.6 <span class="Heading">Steiner Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.7 <span class="Heading">Conjugacy Closed Loops</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.7 <span class="Heading">Small Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.8 <span class="Heading">Small Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.8 <span class="Heading">Paige Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.9 <span class="Heading">Paige Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.9 <span class="Heading">Nilpotent Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.10 <span class="Heading">Nilpotent Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.10 <span class="Heading">Automorphic Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.11 <span class="Heading">Automorphic Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.11 <span class="Heading">Interesting Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.12 <span class="Heading">Interesting Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.13-1 ItpSmallLoop</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapA_mj.html#X7BC4571A79FFB7D0">A <span class="Heading">Files</span></a>

11
doc/chap1.txt
View File

@ -20,7 +20,7 @@
1.2 Installation
Have GAP 4.7 or newer installed on your computer.
Have GAP 4.8 or newer installed on your computer.
If you do not see the subfolder pkg/loops in the main directory of GAP then
download the LOOPS package from the distribution website
@ -85,14 +85,15 @@
We thank the following people for sending us remarks and comments, and for
suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann,
Andreas Distler, Aleš Drápal, Steve Flammia, Kenneth W. Johnson, Michael K.
Kinyon, Alexander Konovalov, Frank Lübeck and Jonathan D.H. Smith.
Andreas Distler, Aleš Drápal, Graham Ellis, Steve Flammia, Kenneth W.
Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck, Jonathan D.H.
Smith, David Stanovský and Glen Whitney.
The library of Moufang loops of order 243 was generated from data provided
by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy
closed loops of order less than 28 was generated from data provided by
Katharina Artic. The library of commutative automorphic loops of order 27,
81 and 243 was obtained jointly with Izabella Stuhl.
Katharina Artic. The library of right Bruck loops of order 27, 81 was
obtained jointly with Izabella Stuhl.
Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr
Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF

6
doc/chap1_mj.html
View File

@ -73,7 +73,7 @@
<h4>1.2 <span class="Heading">Installation</span></h4>
<p>Have <strong class="pkg">GAP 4.7</strong> or newer installed on your computer.</p>
<p>Have <strong class="pkg">GAP 4.8</strong> or newer installed on your computer.</p>
<p>If you do not see the subfolder <code class="file">pkg/loops</code> in the main directory of <strong class="pkg">GAP</strong> then download the <strong class="pkg">LOOPS</strong> package from the distribution website <span class="URL"><a href="http://www.math.du.edu/loops">http://www.math.du.edu/loops</a></span> and unpack the downloaded file into the <code class="file">pkg</code> subfolder.</p>
@ -127,9 +127,9 @@ gap&gt; WriteGapIniFile();;
<h4>1.7 <span class="Heading">Acknowledgment</span></h4>
<p>We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Aleš Drápal, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck and Jonathan D.H. Smith.</p>
<p>We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Aleš Drápal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck, Jonathan D.H. Smith, David Stanovský and Glen Whitney.</p>
<p>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of commutative automorphic loops of order 27, 81 and 243 was obtained jointly with Izabella Stuhl.</p>
<p>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.</p>
<p>Gábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.</p>

4
doc/chap3.txt
View File

@ -40,8 +40,8 @@
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);

44
doc/chap3_mj.html
View File

@ -72,19 +72,19 @@
<pre class="normal">
DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
DeclareCategory( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
DeclareCategory( "IsLoopElement",
IsQuasigroupElement and IsMultiplicativeElementWithInverse );
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
</pre>
@ -162,15 +162,15 @@ DeclareCategory( "IsLoop", IsQuasigroup and
<p>In the following example, <code class="code">L</code> is a loop with two elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L ); </span>
&lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L ); </span>
[ l1, l2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L ); </span>
[ loop_element1, loop_element2 ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print( L );</span>
&lt;loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Elements( L );</span>
[ l1, l2 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( L, "loop_element" );; Elements( L );</span>
[ loop_element1, loop_element2 ]
</pre></div>

140
doc/chap4_mj.html
View File

@ -188,15 +188,15 @@
<p>Since <code class="code">CanonicalCayleyTable</code> is called within the above operation, the resulting quasigroup will have Cayley table with distinct entries <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(\dots\)</span>, <span class="SimpleMath">\(n\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] ); </span>
[ [ 2, 1 ], [ 1, 2 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct ); </span>
[ [ 1, 2 ], [ 2, 1 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last ); </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ]; </span>
[ true, false ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ct := CanonicalCayleyTable( [[5,3],[3,5]] );</span>
[ [ 2, 1 ], [ 1, 2 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NormalizedQuasigroupTable( ct );</span>
[ [ 1, 2 ], [ 2, 1 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByCayleyTable( last );</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsQuasigroupTable( ct ), IsLoopTable( ct ) ];</span>
[ true, false ]
</pre></div>
<p><a id="X849944F17E2B37F8" name="X849944F17E2B37F8"></a></p>
@ -235,56 +235,56 @@
<p><strong class="button">Example:</strong> Data does not have to be arranged into an array of any kind.</p>
<p class="center">\[
\begin{array}{cccc}
0&amp;1&amp;2&amp;1\\
2&amp;0&amp;2&amp; \\
0&amp;1&amp; &amp;
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\\
2&amp;3&amp;1\\
3&amp;1&amp;2
\end{array}
<p class="center">\[
\begin{array}{cccc}
0&amp;1&amp;2&amp;1\\
2&amp;0&amp;2&amp; \\
0&amp;1&amp; &amp;
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\\
2&amp;3&amp;1\\
3&amp;1&amp;2
\end{array}
\]</p>
<p><strong class="button">Example:</strong> Chunks can be any strings.</p>
<p class="center">\[
\begin{array}{cc}
{\rm red}&amp;{\rm green}\\
{\rm green}&amp;{\rm red}\\
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
<p class="center">\[
\begin{array}{cc}
{\rm red}&amp;{\rm green}\\
{\rm green}&amp;{\rm red}\\
\end{array}\quad + \quad "" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
\]</p>
<p><strong class="button">Example:</strong> A typical table produced by <strong class="pkg">GAP</strong> is easily parsed by deleting brackets and commas.</p>
<p class="center">\[
[ [0, 1], [1, 0] ] \quad + \quad "[,]" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
<p class="center">\[
[ [0, 1], [1, 0] ] \quad + \quad "[,]" \quad \Longrightarrow\quad
\begin{array}{cc}
1&amp; 2\\
2&amp; 1
\end{array}
\]</p>
<p><strong class="button">Example:</strong> A typical TeX table with rows separated by lines is also easily converted. Note that we have to use <span class="SimpleMath">\(\backslash\backslash\)</span> to ensure that every occurrence of <span class="SimpleMath">\(\backslash\)</span> is deleted, since <span class="SimpleMath">\(\backslash\backslash\)</span> represents the character <span class="SimpleMath">\(\backslash\)</span> in <strong class="pkg">GAP</strong></p>
<p class="center">\[
\begin{array}{lll}
x\&amp;&amp; y\&amp;&amp;\ z\backslash\backslash\cr
y\&amp;&amp; z\&amp;&amp;\ x\backslash\backslash\cr
z\&amp;&amp; x\&amp;&amp;\ y
\end{array}
\quad + \quad "\backslash\backslash\&amp;" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\cr
2&amp;3&amp;1\cr
3&amp;1&amp;2
\end{array}
<p class="center">\[
\begin{array}{lll}
x\&amp;&amp; y\&amp;&amp;\ z\backslash\backslash\cr
y\&amp;&amp; z\&amp;&amp;\ x\backslash\backslash\cr
z\&amp;&amp; x\&amp;&amp;\ y
\end{array}
\quad + \quad "\backslash\backslash\&amp;" \quad \Longrightarrow\quad
\begin{array}{ccc}
1&amp;2&amp;3\cr
2&amp;3&amp;1\cr
3&amp;1&amp;2
\end{array}
\]</p>
<p><a id="X81A1DB918057933E" name="X81A1DB918057933E"></a></p>
@ -329,14 +329,14 @@
<p>These are the dual operations to <code class="code">QuasigroupByLeftSection</code> and <code class="code">LoopByLeftSection</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">S := Subloop( MoufangLoop( 12, 1 ), [ 3 ] );; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ls := LeftSection( S ); </span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls ); </span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) ); </span>
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">S := Subloop( MoufangLoop( 12, 1 ), [ 3 ] );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ls := LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTableByPerms( ls );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( LoopByLeftSection( ls ) );</span>
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 3, 1, 2 ] ]
</pre></div>
<p><a id="X85ABE99E84E5B0E8" name="X85ABE99E84E5B0E8"></a></p>
@ -360,11 +360,11 @@
<p>Here is a simple example in which <span class="SimpleMath">\(T\)</span> is actually the right section of the resulting loop.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">T := [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ];; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := Group( T );; H := Stabilizer( G, 1 );; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T ); </span>
&lt;loop of order 5&gt;
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">T := [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">G := Group( T );; H := Stabilizer( G, 1 );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByRightFolder( G, H, T );</span>
&lt;loop of order 5&gt;
</pre></div>
<p><a id="X8759431780AC81A9" name="X8759431780AC81A9"></a></p>
@ -394,16 +394,16 @@
<p>Returns: The extension of an abelian group <var class="Arg">K</var> by a loop <var class="Arg">F</var>, using action <var class="Arg">f</var> and cocycle <var class="Arg">t</var>. The arguments must be formatted as the output of <code class="code">NuclearExtension</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) ); </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">K := DirectProduct( F, F );; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">phi := [ (), (2,3) ];; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">theta := [ [ 1, 1 ], [ 1, 3 ] ];; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByExtension( K, F, phi, theta ); </span>
&lt;loop of order 8&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last ); </span>
false
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := IntoLoop( Group( (1,2) ) );</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">K := DirectProduct( F, F );;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">phi := [ (), (2,3) ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">theta := [ [ 1, 1 ], [ 1, 3 ] ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LoopByExtension( K, F, phi, theta );</span>
&lt;loop of order 8&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsAssociative( last );</span>
false
</pre></div>
<p><a id="X7AE29A1A7AA5C25A" name="X7AE29A1A7AA5C25A"></a></p>

22
doc/chap5_mj.html
View File

@ -167,17 +167,17 @@
<p>Returns: The left inverse, right inverse and inverse, respectively, of the quasigroup element <var class="Arg">x</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q ); </span>
[ [ 1, 2, 3, 4, 5 ],
[ 2, 1, 4, 5, 3 ],
[ 3, 4, 5, 1, 2 ],
[ 4, 5, 2, 3, 1 ],
[ 5, 3, 1, 2, 4 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">elms := Elements( Q ); </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ l1, l2, l3, l4, l5 ]; </span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ]; </span>
[ l5, l4, fail ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( Q );</span>
[ [ 1, 2, 3, 4, 5 ],
[ 2, 1, 4, 5, 3 ],
[ 3, 4, 5, 1, 2 ],
[ 4, 5, 2, 3, 1 ],
[ 5, 3, 1, 2, 4 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">elms := Elements( Q );</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ l1, l2, l3, l4, l5 ];</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ];</span>
[ l5, l4, fail ]
</pre></div>
<p><a id="X7E0849977869E53D" name="X7E0849977869E53D"></a></p>

31
doc/chap6.txt
View File

@ -488,17 +488,28 @@
the elements of the underlying quasigroup without changing the isomorphism
type of the quasigroups. LOOPS contains several functions for this purpose.
6.11-6 IsomorphicCopyByPerm
6.11-6 QuasigroupIsomorph
QuasigroupIsomorph( Q, f )  operation
Returns: When Q is a quasigroup and f is a permutation of 1,dots,|Q|,
returns the quasigroup defined on the same set as Q with
multiplication * defined by x*y =f(f^-1(x)f^-1(y)).
6.11-7 LoopIsomorph
LoopIsomorph( Q, f )  operation
Returns: When Q is a loop and f is a permutation of 1,dots,|Q| fixing 1,
returns the loop defined on the same set as Q with multiplication
* defined by x*y =f(f^-1(x)f^-1(y)). If f(1)=cne 1, the
isomorphism (1,c) is applied after f.
6.11-8 IsomorphicCopyByPerm
IsomorphicCopyByPerm( Q, f )  operation
Returns: When Q is a quasigroup and f is a permutation of 1,dots,|Q|,
returns a quasigroup defined on the same set as Q with
multiplication * defined by x*y =f(f^-1(x)f^-1(y)). When Q is a
declared loop, a loop is returned. Consequently, when Q is a
declared loop and f(1) = kne 1, then f is first replaced with f∘
(1,k), to make sure that the resulting Cayley table is normalized.
Returns: LoopIsomorphism(Q,f) if Q is a loop, and
QuasigroupIsomorphism(Q,f) if Q is a quasigroup.
6.11-7 IsomorphicCopyByNormalSubloop
6.11-9 IsomorphicCopyByNormalSubloop
IsomorphicCopyByNormalSubloop( Q, S )  operation
Returns: When S is a normal subloop of a loop Q, returns an isomorphic copy
@ -511,7 +522,7 @@
these invariants to partition the loop into blocks of elements preserved
under isomorphisms. The following two operations are used in the search.
6.11-8 Discriminator
6.11-10 Discriminator
Discriminator( Q )  operation
Returns: A data structure with isomorphism invariants of a loop Q.
@ -523,7 +534,7 @@
If two loops have different discriminators, they are not isomorphic. If they
have identical discriminators, they may or may not be isomorphic.
6.11-9 AreEqualDiscriminators
6.11-11 AreEqualDiscriminators
AreEqualDiscriminators( D1, D2 )  operation
Returns: true if D1, D2 are equal discriminators for the purposes of

118
doc/chap6_mj.html
View File

@ -120,10 +120,12 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
</span>
@ -265,28 +267,28 @@
<p>Note how the Cayley table of a subquasigroup is created only upon explicit demand. Also note that changing the names of elements of a subquasigroup (subloop) automatically changes the names of the elements of the parent subquasigroup (subloop). This is because the elements are shared.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] ); </span>
&lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ]; </span>
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S ); </span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M ); </span>
[ s1, s3, s5 ]
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S ); </span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S ); </span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ]; </span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L ); </span>
[ l1, l2, l3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ]; </span>
[ true, true, false ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L ); </span>
[ (), (1,2,3), (1,3,2) ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.5 ] );</span>
&lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( S ) = M, Elements( S ), PosInParent( S ) ];</span>
[ true, [ l1, l3, l5], [ 1, 3, 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasCayleyTable( S );</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetLoopElmName( S, "s" );; Elements( S ); Elements( M );</span>
[ s1, s3, s5 ]
[ s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CayleyTable( S );</span>
[ [ 1, 3, 5 ], [ 3, 5, 1 ], [ 5, 1, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( S );</span>
[ (), (1,3,5), (1,5,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasCayleyTable( S ), Parent( S ) = M ];</span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( CayleyTable( S ) );; Elements( L );</span>
[ l1, l2, l3 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ Parent( L ) = L, IsSubloop( M, S ), IsSubloop( M, L ) ];</span>
[ true, true, false ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection( L );</span>
[ (), (1,2,3), (1,3,2) ]
</pre></div>
<p><a id="X78ED50F578A88046" name="X78ED50F578A88046"></a></p>
@ -349,16 +351,16 @@ false
<p>Here is an example for multiplication groups and inner mapping groups:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1); </span>
&lt;Moufang loop 12/1&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection(M)[2]; </span>
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M); </span>
&lt;permutation group of size 2592 with 23 generators&gt;
Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Inn); </span>
216
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop(12,1);</span>
&lt;Moufang loop 12/1&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LeftSection(M)[2];</span>
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Mlt := MultiplicationGroup(M); Inn := InnerMappingGroup(M);</span>
&lt;permutation group of size 2592 with 23 generators&gt;
Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Inn);</span>
216
</pre></div>
<p><a id="X7B45C2AF7C2E28AB" name="X7B45C2AF7C2E28AB"></a></p>
@ -464,16 +466,16 @@ Group([ (4,6)(7,11), (7,11)(8,10), (2,6,4)(7,9,11), (3,5)(9,11), (8,12,10) ])
<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns the natural projection from <var class="Arg">Q</var> onto <var class="Arg">Q</var><span class="SimpleMath">\(/\)</span><var class="Arg">S</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] ); </span>
&lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S ); </span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S ); </span>
&lt;loop of order 4&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S ); </span>
MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
function( x ) ... end )
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">M := MoufangLoop( 12, 1 );; S := Subloop( M, [ M.3 ] );</span>
&lt;loop of order 3&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsNormal( M, S );</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">F := FactorLoop( M, S );</span>
&lt;loop of order 4&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">NaturalHomomorphismByNormalSubloop( M, S );</span>
MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
function( x ) ... end )
</pre></div>
<p><a id="X821F40748401D698" name="X821F40748401D698"></a></p>
@ -607,16 +609,30 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <strong class="pkg">LOOPS</strong> contains several functions for this purpose.</p>
<p><a id="X7A42812B7B027DD4" name="X7A42812B7B027DD4"></a></p>
<h5>6.11-6 QuasigroupIsomorph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; QuasigroupIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns the quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>.</p>
<p><a id="X7BD1AC32851286EA" name="X7BD1AC32851286EA"></a></p>
<h5>6.11-7 LoopIsomorph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LoopIsomorph</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a loop and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span> fixing <span class="SimpleMath">\(1\)</span>, returns the loop defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. If <var class="Arg">f</var><span class="SimpleMath">\((1)=c\ne 1\)</span>, the isomorphism <span class="SimpleMath">\((1,c)\)</span> is applied after <var class="Arg">f</var>.</p>
<p><a id="X85B3E22679FD8D81" name="X85B3E22679FD8D81"></a></p>
<h5>6.11-6 IsomorphicCopyByPerm</h5>
<h5>6.11-8 IsomorphicCopyByPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByPerm</code>( <var class="Arg">Q</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">Q</var> is a quasigroup and <var class="Arg">f</var> is a permutation of <span class="SimpleMath">\(1,\dots,|\)</span><var class="Arg">Q</var><span class="SimpleMath">\(|\)</span>, returns a quasigroup defined on the same set as <var class="Arg">Q</var> with multiplication <span class="SimpleMath">\(*\)</span> defined by <span class="SimpleMath">\(x*y = \)</span><var class="Arg">f</var><span class="SimpleMath">\((\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(x)\)</span><var class="Arg">f</var><span class="SimpleMath">\({}^{-1}(y))\)</span>. When <var class="Arg">Q</var> is a declared loop, a loop is returned. Consequently, when <var class="Arg">Q</var> is a declared loop and <var class="Arg">f</var><span class="SimpleMath">\((1) = k\ne 1\)</span>, then <var class="Arg">f</var> is first replaced with <var class="Arg">f</var><span class="SimpleMath">\(\circ (1,k)\)</span>, to make sure that the resulting Cayley table is normalized.</p>
<p>Returns: <code class="code">LoopIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a loop, and <code class="code">QuasigroupIsomorphism(<var class="Arg">Q</var>,<var class="Arg">f</var>)</code> if <var class="Arg">Q</var> is a quasigroup.</p>
<p><a id="X8121DE3A78795040" name="X8121DE3A78795040"></a></p>
<h5>6.11-7 IsomorphicCopyByNormalSubloop</h5>
<h5>6.11-9 IsomorphicCopyByNormalSubloop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsomorphicCopyByNormalSubloop</code>( <var class="Arg">Q</var>, <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: When <var class="Arg">S</var> is a normal subloop of a loop <var class="Arg">Q</var>, returns an isomorphic copy of <var class="Arg">Q</var> in which the elements are ordered according to the right cosets of <var class="Arg">S</var>. In particular, the Cayley table of <var class="Arg">S</var> will appear in the top left corner of the Cayley table of the resulting loop.<br /></p>
@ -625,7 +641,7 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p><a id="X7D09D8957E4A0973" name="X7D09D8957E4A0973"></a></p>
<h5>6.11-8 Discriminator</h5>
<h5>6.11-10 Discriminator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Discriminator</code>( <var class="Arg">Q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A data structure with isomorphism invariants of a loop <var class="Arg">Q</var>.</p>
@ -636,7 +652,7 @@ MappingByFunction( &lt;loop of order 12&gt;, &lt;loop of order 4&gt;,
<p><a id="X812F0DEE7C896E18" name="X812F0DEE7C896E18"></a></p>
<h5>6.11-9 AreEqualDiscriminators</h5>
<h5>6.11-11 AreEqualDiscriminators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AreEqualDiscriminators</code>( <var class="Arg">D1</var>, <var class="Arg">D2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="code">true</code> if <var class="Arg">D1</var>, <var class="Arg">D2</var> are equal discriminators for the purposes of isomorphism searches.</p>

26
doc/chap7_mj.html
View File

@ -444,19 +444,19 @@
<p>The following trivial example shows some of the implications and the naming conventions of <strong class="pkg">LOOPS</strong> at work:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] ); </span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ] </span>
[ true, &lt;left Bol loop of order 2&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ]; </span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ]; </span>
[ false, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L; </span>
&lt;Moufang loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ]; </span>
[ true, &lt;associative loop of order 2&gt; ]
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L := LoopByCayleyTable( [ [ 1, 2 ], [ 2, 1 ] ] );</span>
&lt;loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsLeftBolLoop( L ), L ]</span>
[ true, &lt;left Bol loop of order 2&gt; ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsLeftAlternativeLoop( L ), IsLeftAlternativeLoop( L ) ];</span>
[ true, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ HasIsRightBolLoop( L ), IsRightBolLoop( L ) ];</span>
[ false, true ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">L;</span>
&lt;Moufang loop of order 2&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">[ IsAssociative( L ), L ];</span>
[ true, &lt;associative loop of order 2&gt; ]
</pre></div>
<p>The analogous terminology for quasigroups of Bol-Moufang type is not standard yet, and hence is not supported in <strong class="pkg">LOOPS</strong> except for the situations explicitly noted above.</p>

14
doc/chap8_mj.html
View File

@ -202,13 +202,13 @@
<p>Returns: One loop (given as a section) whose multiplication group is equal to the transitive permutation group <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=PGL(3,3); </span>
Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a); </span>
56
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a); </span>
52
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=PGL(3,3);</span>
Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopTablesInGroup(g,3,0);; Size(a);</span>
56
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AllLoopsWithMltGroup(g,3,0);; Size(a);</span>
52
</pre></div>

86
doc/chap9.txt
View File

@ -78,12 +78,34 @@
retrieved by calling Opposite on left Bol loops.
9.3 Moufang Loops
9.3 Left Bruck Loops and Right Bruck Loops
The emmerging library named left Bruck contains all left Bruck loops of
orders 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively).
For an odd prime p, left Bruck loops of order p^k are centrally nilpotent
and hence central extensions of the cyclic group of order p by a left Bruck
loop of order p^k-1. It is known that left Bruck loops of order p and p^2
are abelian groups; we have included them in the library because of the
iterative nature of the construction of nilpotent loops.
9.3-1 LeftBruckLoop
LeftBruckLoop( n, m )  function
Returns: The mth left Bruck loop of order n in the library.
9.3-2 RightBruckLoop
RightBruckLoop( n, m )  function
Returns: The mth right Bruck loop of order n in the library.
9.4 Moufang Loops
The library named Moufang contains all nonassociative Moufang loops of order
nle 64 and n∈{81,243}.
9.3-1 MoufangLoop
9.4-1 MoufangLoop
MoufangLoop( n, m )  function
Returns: The mth Moufang loop of order n in the library.
@ -107,20 +129,20 @@
obtained as MoufangLoop(16,3).
9.4 Code Loops
9.5 Code Loops
The library named code contains all nonassociative code loops of order less
than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order
64, all Moufang. The library merely points to the corresponding Moufang
loops. See [NV07] for a classification of small code loops.
9.4-1 CodeLoop
9.5-1 CodeLoop
CodeLoop( n, m )  function
Returns: The mth code loop of order n in the library.
9.5 Steiner Loops
9.6 Steiner Loops
Here is how the libary named Steiner is described within LOOPS:
@ -141,13 +163,13 @@
Our labeling of Steiner loops of order 16 coincides with the labeling of
Steiner triple systems of order 15 in [CR99].
9.5-1 SteinerLoop
9.6-1 SteinerLoop
SteinerLoop( n, m )  function
Returns: The mth Steiner loop of order n in the library.
9.6 Conjugacy Closed Loops
9.7 Conjugacy Closed Loops
The library named RCC contains all nonassocitive right conjugacy closed
loops of order nle 27 up to isomorphism. The data for the library was
@ -171,14 +193,14 @@
9.6-1 RCCLoop and RightConjugacyClosedLoop
9.7-1 RCCLoop and RightConjugacyClosedLoop
RCCLoop( n, m )  function
RightConjugacyClosedLoop( n, m )  function
Returns: The mth right conjugacy closed loop of order n in the library.
9.6-2 LCCLoop and LeftConjugacyClosedLoop
9.7-2 LCCLoop and LeftConjugacyClosedLoop
LCCLoop( n, m )  function
LeftConjugacyClosedLoop( n, m )  function
@ -188,8 +210,10 @@
Left conjugacy closed loops are obtained from right conjugacy closed loops
via Opposite.
The library named CC contains all nonassociative conjugacy closed loops of
order nle 27 and also of orders 2p and p^2 for all primes p.
The library named CC contains all CC loops of order 2le 2^kle 64, 3le 3^kle
81, 5le 5^kle 125, 7le 7^kle 343, all nonassociative CC loops of order less
than 28, and all nonassociative CC loops of order p^2 and 2p for any odd
prime p.
By results of Kunen [Kun00], for every odd prime p there are precisely 3
nonassociative conjugacy closed loops of order p^2. Csörgő and Drápal [CD05]
@ -215,25 +239,25 @@
m + n ).
9.6-3 CCLoop and ConjugacyClosedLoop
9.7-3 CCLoop and ConjugacyClosedLoop
CCLoop( n, m )  function
ConjugacyClosedLoop( n, m )  function
Returns: The mth conjugacy closed loop of order n in the library.
9.7 Small Loops
9.8 Small Loops
The library named small contains all nonassociative loops of order 5 and 6.
There are 5 and 107 such loops, respectively.
9.7-1 SmallLoop
9.8-1 SmallLoop
SmallLoop( n, m )  function
Returns: The mth loop of order n in the library.
9.8 Paige Loops
9.9 Paige Loops
Paige loops are nonassociative finite simple Moufang loops. By [Lie87],
there is precisely one Paige loop for every finite field.
@ -241,14 +265,14 @@
The library named Paige contains the smallest nonassociative simple Moufang
loop.
9.8-1 PaigeLoop
9.9-1 PaigeLoop
PaigeLoop( q )  function
Returns: The Paige loop constructed over the finite field of order q. Only
the case q=2 is implemented.
9.9 Nilpotent Loops
9.10 Nilpotent Loops
The library named nilpotent contains all nonassociative nilpotent loops of
order less than 12 up to isomorphism. There are 2 nonassociative nilpotent
@ -258,30 +282,32 @@
are 2623755 nilpotent loops of order 12, and 123794003928541545927226368
nilpotent loops of order 22.
9.9-1 NilpotentLoop
9.10-1 NilpotentLoop
NilpotentLoop( n, m )  function
Returns: The mth nilpotent loop of order n in the library.
9.10 Automorphic Loops
9.11 Automorphic Loops
The library named automorphic contains all nonassociative automorphic loops
of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7
of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15),
all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2,
7 and 72 such loops, respectively, including abelian groups), and
commutative automorphic loops Q of order 243 possessing a central subloop S
of order 3 such that Q/S is not the elementary abelian group of order 81
(there are 118451 such loops).
of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15)
and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1,
2, 7 and 72 such loops).
9.10-1 AutomorphicLoop
It turns out that commutative automorphic loops of order 3, 9, 27 and 81
(but not 243) are in one-to-on correspondence with left Bruck loops of the
respective orders, see [Gre14], [SV17]. Only the left Bruck loops are stored
in the library.
9.11-1 AutomorphicLoop
AutomorphicLoop( n, m )  function
Returns: The mth automorphic loop of order n in the library.
9.11 Interesting Loops
9.12 Interesting Loops
The library named interesting contains some loops that are illustrative in
the theory of loops. At this point, the library contains a nonassociative
@ -290,20 +316,20 @@
generalize octonions), and the unique nonassociative simple right Bol loop
of order 96 and exponent 2.
9.11-1 InterestingLoop
9.12-1 InterestingLoop
InterestingLoop( n, m )  function
Returns: The mth interesting loop of order n in the library.
9.12 Libraries of Loops Up To Isotopism
9.13 Libraries of Loops Up To Isotopism
For the library named small we also provide the corresponding library of
loops up to isotopism. In general, given a library named libname, the
corresponding library of loops up to isotopism is named itp lib, and the
loops can be retrieved by the template ItpLibLoop(n,m).
9.12-1 ItpSmallLoop
9.13-1 ItpSmallLoop
ItpSmallLoop( n, m )  function
Returns: The mth small loop of order n up to isotopism in the library.

186
doc/chap9_mj.html
View File

@ -38,60 +38,66 @@
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8028D69A86B15897">9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7953702D84E60AF4">9.4 <span class="Heading">Moufang Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.5 <span class="Heading">Steiner Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.5 <span class="Heading">Code Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.6 <span class="Heading">Conjugacy Closed Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X84E941EE7846D3EE">9.6 <span class="Heading">Steiner Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X867E5F0783FEB8B5">9.7 <span class="Heading">Conjugacy Closed Loops</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X806B2DE67990E42F">9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X798BC601843E8916">9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.7 <span class="Heading">Small Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X7E3A8F2C790F2CA1">9.8 <span class="Heading">Small Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.8 <span class="Heading">Paige Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X8135C8FD8714C606">9.9 <span class="Heading">Paige Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.9 <span class="Heading">Nilpotent Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X86695C577A4D1784">9.10 <span class="Heading">Nilpotent Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.10 <span class="Heading">Automorphic Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X793B22EA8643C667">9.11 <span class="Heading">Automorphic Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.11 <span class="Heading">Interesting Loops</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X843BD73F788049F7">9.12 <span class="Heading">Interesting Loops</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap9_mj.html#X864839227D5C0A90">9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap9_mj.html#X850C4C01817A098D">9.13-1 ItpSmallLoop</a></span>
</div></div>
</div>
@ -170,15 +176,37 @@
<p><strong class="button">Remark:</strong> Only left Bol loops are stored in the library. Right Bol loops are retrieved by calling <code class="code">Opposite</code> on left Bol loops.</p>
<p><a id="X8028D69A86B15897" name="X8028D69A86B15897"></a></p>
<h4>9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></h4>
<p>The emmerging library named <em>left Bruck</em> contains all left Bruck loops of orders <span class="SimpleMath">\(3\)</span>, <span class="SimpleMath">\(9\)</span>, <span class="SimpleMath">\(27\)</span> and <span class="SimpleMath">\(81\)</span> (there are <span class="SimpleMath">\(1\)</span>, <span class="SimpleMath">\(2\)</span>, <span class="SimpleMath">\(7\)</span> and <span class="SimpleMath">\(72\)</span> such loops, respectively).</p>
<p>For an odd prime <span class="SimpleMath">\(p\)</span>, left Bruck loops of order <span class="SimpleMath">\(p^k\)</span> are centrally nilpotent and hence central extensions of the cyclic group of order <span class="SimpleMath">\(p\)</span> by a left Bruck loop of order <span class="SimpleMath">\(p^{k-1}\)</span>. It is known that left Bruck loops of order <span class="SimpleMath">\(p\)</span> and <span class="SimpleMath">\(p^2\)</span> are abelian groups; we have included them in the library because of the iterative nature of the construction of nilpotent loops.</p>
<p><a id="X8290B01780F0FCD3" name="X8290B01780F0FCD3"></a></p>
<h5>9.3-1 LeftBruckLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftBruckLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th left Bruck loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X798DD7CF871F648F" name="X798DD7CF871F648F"></a></p>
<h5>9.3-2 RightBruckLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightBruckLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th right Bruck loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X7953702D84E60AF4" name="X7953702D84E60AF4"></a></p>
<h4>9.3 <span class="Heading">Moufang Loops</span></h4>
<h4>9.4 <span class="Heading">Moufang Loops</span></h4>
<p>The library named <em>Moufang</em> contains all nonassociative Moufang loops of order <span class="SimpleMath">\(n\le 64\)</span> and <span class="SimpleMath">\(n\in\{81,243\}\)</span>.</p>
<p><a id="X81E82098822543EE" name="X81E82098822543EE"></a></p>
<h5>9.3-1 MoufangLoop</h5>
<h5>9.4-1 MoufangLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MoufangLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th Moufang loop of order <var class="Arg">n</var> in the library.</p>
@ -187,79 +215,79 @@
<p>The extent of the library is summarized below:</p>
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrrrr}
order&amp;12&amp;16&amp;20&amp;24&amp;28&amp;32&amp;36&amp;40&amp;42&amp;44&amp;48&amp;52&amp;54&amp;56&amp;60&amp;64&amp;81&amp;243\cr
loops&amp;1 &amp;5 &amp;1 &amp;5 &amp;1 &amp;71&amp;4 &amp;5 &amp;1 &amp;1 &amp;51&amp;1 &amp;2 &amp;4 &amp;5 &amp;4262&amp; 5 &amp;72
\end{array}
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrrrr}
order&amp;12&amp;16&amp;20&amp;24&amp;28&amp;32&amp;36&amp;40&amp;42&amp;44&amp;48&amp;52&amp;54&amp;56&amp;60&amp;64&amp;81&amp;243\cr
loops&amp;1 &amp;5 &amp;1 &amp;5 &amp;1 &amp;71&amp;4 &amp;5 &amp;1 &amp;1 &amp;51&amp;1 &amp;2 &amp;4 &amp;5 &amp;4262&amp; 5 &amp;72
\end{array}
\]</p>
<p>The <em>octonion loop</em> of order 16 (i.e., the multiplication loop of the basis elements in the 8-dimensional standard real octonion algebra) can be obtained as <code class="code">MoufangLoop(16,3)</code>.</p>
<p><a id="X7BCA6BCB847F79DC" name="X7BCA6BCB847F79DC"></a></p>
<h4>9.4 <span class="Heading">Code Loops</span></h4>
<h4>9.5 <span class="Heading">Code Loops</span></h4>
<p>The library named <em>code</em> contains all nonassociative code loops of order less than 65. There are 5 such loops of order 16, 16 of order 32, and 80 of order 64, all Moufang. The library merely points to the corresponding Moufang loops. See <a href="chapBib_mj.html#biBNaVo2007">[NV07]</a> for a classification of small code loops.</p>
<p><a id="X7DB4D3B27BB4D7EE" name="X7DB4D3B27BB4D7EE"></a></p>
<h5>9.4-1 CodeLoop</h5>
<h5>9.5-1 CodeLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CodeLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th code loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X84E941EE7846D3EE" name="X84E941EE7846D3EE"></a></p>
<h4>9.5 <span class="Heading">Steiner Loops</span></h4>
<h4>9.6 <span class="Heading">Steiner Loops</span></h4>
<p>Here is how the libary named <em>Steiner</em> is described within <strong class="pkg">LOOPS</strong>:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DisplayLibraryInfo( "Steiner" ); </span>
The library contains all nonassociative Steiner loops of order less or equal to 16.
It also contains the associative Steiner loops of order 4 and 8.
------
Extent of the library:
1 loop of order 4
1 loop of order 8
1 loop of order 10
2 loops of order 14
80 loops of order 16
true
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">DisplayLibraryInfo( "Steiner" );</span>
The library contains all nonassociative Steiner loops of order less or equal to 16.
It also contains the associative Steiner loops of order 4 and 8.
------
Extent of the library:
1 loop of order 4
1 loop of order 8
1 loop of order 10
2 loops of order 14
80 loops of order 16
true
</pre></div>
<p>Our labeling of Steiner loops of order 16 coincides with the labeling of Steiner triple systems of order 15 in <a href="chapBib_mj.html#biBCoRo">[CR99]</a>.</p>
<p><a id="X87C235457E859AF4" name="X87C235457E859AF4"></a></p>
<h5>9.5-1 SteinerLoop</h5>
<h5>9.6-1 SteinerLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SteinerLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th Steiner loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X867E5F0783FEB8B5" name="X867E5F0783FEB8B5"></a></p>
<h4>9.6 <span class="Heading">Conjugacy Closed Loops</span></h4>
<h4>9.7 <span class="Heading">Conjugacy Closed Loops</span></h4>
<p>The library named <em>RCC</em> contains all nonassocitive right conjugacy closed loops of order <span class="SimpleMath">\(n\le 27\)</span> up to isomorphism. The data for the library was generated by Katharina Artic <a href="chapBib_mj.html#biBArtic">[Art15]</a> who can also provide additional data for all right conjugacy closed loops of order <span class="SimpleMath">\(n\le 31\)</span>.</p>
<p>Let <span class="SimpleMath">\(Q\)</span> be a right conjugacy closed loop, <span class="SimpleMath">\(G\)</span> its right multiplication group and <span class="SimpleMath">\(T\)</span> its right section. Then <span class="SimpleMath">\(\langle T\rangle = G\)</span> is a transitive group, and <span class="SimpleMath">\(T\)</span> is a union of conjugacy classes of <span class="SimpleMath">\(G\)</span>. Every right conjugacy closed loop of order <span class="SimpleMath">\(n\)</span> can therefore be represented as a union of certain conjugacy classes of a transitive group of degree <span class="SimpleMath">\(n\)</span>. This is how right conjugacy closed loops of order less than <span class="SimpleMath">\(28\)</span> are represented in <strong class="pkg">LOOPS</strong>. The following table summarizes the number of right conjugacy closed loops of a given order up to isomorphism:</p>
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrr}
order &amp;6&amp; 8&amp;9&amp;10&amp; 12&amp;14&amp;15&amp; 16&amp; 18&amp; 20&amp;\cr
loops &amp;3&amp;19&amp;5&amp;16&amp;155&amp;97&amp; 17&amp;6317&amp;1901&amp;8248&amp;\cr
\hline
order &amp;21&amp; 22&amp; 24&amp; 25&amp; 26&amp; 27\cr
loops &amp;119&amp;10487&amp;471995&amp; 119&amp;151971&amp;152701
\end{array}
<p class="center">\[
\begin{array}{r|rrrrrrrrrrrrrrrr}
order &amp;6&amp; 8&amp;9&amp;10&amp; 12&amp;14&amp;15&amp; 16&amp; 18&amp; 20&amp;\cr
loops &amp;3&amp;19&amp;5&amp;16&amp;155&amp;97&amp; 17&amp;6317&amp;1901&amp;8248&amp;\cr
\hline
order &amp;21&amp; 22&amp; 24&amp; 25&amp; 26&amp; 27\cr
loops &amp;119&amp;10487&amp;471995&amp; 119&amp;151971&amp;152701
\end{array}
\]</p>
<p><a id="X806B2DE67990E42F" name="X806B2DE67990E42F"></a></p>
<h5>9.6-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></h5>
<h5>9.7-1 <span class="Heading">RCCLoop and RightConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RightConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -267,7 +295,7 @@ true
<p><a id="X80AB8B107D55FB19" name="X80AB8B107D55FB19"></a></p>
<h5>9.6-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></h5>
<h5>9.7-2 <span class="Heading">LCCLoop and LeftConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LCCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; LeftConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -275,7 +303,7 @@ true
<p><strong class="button">Remark:</strong> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <code class="code">Opposite</code>.<br /></p>
<p>The library named <em>CC</em> contains all nonassociative conjugacy closed loops of order <span class="SimpleMath">\(n\le 27\)</span> and also of orders <span class="SimpleMath">\(2p\)</span> and <span class="SimpleMath">\(p^2\)</span> for all primes <span class="SimpleMath">\(p\)</span>.</p>
<p>The library named <em>CC</em> contains all CC loops of order <span class="SimpleMath">\(2\le 2^k\le 64\)</span>, <span class="SimpleMath">\(3\le 3^k\le 81\)</span>, <span class="SimpleMath">\(5\le 5^k\le 125\)</span>, <span class="SimpleMath">\(7\le 7^k\le 343\)</span>, all nonassociative CC loops of order less than 28, and all nonassociative CC loops of order <span class="SimpleMath">\(p^2\)</span> and <span class="SimpleMath">\(2p\)</span> for any odd prime <span class="SimpleMath">\(p\)</span>.</p>
<p>By results of Kunen <a href="chapBib_mj.html#biBKun">[Kun00]</a>, for every odd prime <span class="SimpleMath">\(p\)</span> there are precisely 3 nonassociative conjugacy closed loops of order <span class="SimpleMath">\(p^2\)</span>. Csörgő and Drápal <a href="chapBib_mj.html#biBCsDr">[CD05]</a> described these 3 loops by multiplicative formulas on <span class="SimpleMath">\(\mathbb{Z}_{p^2}\)</span> and <span class="SimpleMath">\(\mathbb{Z}_p \times \mathbb{Z}_p\)</span> as follows:</p>
@ -295,7 +323,7 @@ true
<p><a id="X798BC601843E8916" name="X798BC601843E8916"></a></p>
<h5>9.6-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></h5>
<h5>9.7-3 <span class="Heading">CCLoop and ConjugacyClosedLoop</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CCLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ConjugacyClosedLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
@ -303,20 +331,20 @@ true
<p><a id="X7E3A8F2C790F2CA1" name="X7E3A8F2C790F2CA1"></a></p>
<h4>9.7 <span class="Heading">Small Loops</span></h4>
<h4>9.8 <span class="Heading">Small Loops</span></h4>
<p>The library named <em>small</em> contains all nonassociative loops of order 5 and 6. There are 5 and 107 such loops, respectively.</p>
<p><a id="X7C6EE23E84CD87D3" name="X7C6EE23E84CD87D3"></a></p>
<h5>9.7-1 SmallLoop</h5>
<h5>9.8-1 SmallLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X8135C8FD8714C606" name="X8135C8FD8714C606"></a></p>
<h4>9.8 <span class="Heading">Paige Loops</span></h4>
<h4>9.9 <span class="Heading">Paige Loops</span></h4>
<p><em>Paige loops</em> are nonassociative finite simple Moufang loops. By <a href="chapBib_mj.html#biBLi">[Lie87]</a>, there is precisely one Paige loop for every finite field.</p>
@ -324,14 +352,14 @@ true
<p><a id="X7FCF4D6B7AD66D74" name="X7FCF4D6B7AD66D74"></a></p>
<h5>9.8-1 PaigeLoop</h5>
<h5>9.9-1 PaigeLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PaigeLoop</code>( <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The Paige loop constructed over the finite field of order <var class="Arg">q</var>. Only the case <code class="code"><var class="Arg">q</var>=2</code> is implemented.</p>
<p><a id="X86695C577A4D1784" name="X86695C577A4D1784"></a></p>
<h4>9.9 <span class="Heading">Nilpotent Loops</span></h4>
<h4>9.10 <span class="Heading">Nilpotent Loops</span></h4>
<p>The library named <em>nilpotent</em> contains all nonassociative nilpotent loops of order less than 12 up to isomorphism. There are 2 nonassociative nilpotent loops of order 6, 134 of order 8, 8 of order 9 and 1043 of order 10.</p>
@ -339,58 +367,60 @@ true
<p><a id="X7A9C960D86E2AD28" name="X7A9C960D86E2AD28"></a></p>
<h5>9.9-1 NilpotentLoop</h5>
<h5>9.10-1 NilpotentLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; NilpotentLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th nilpotent loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X793B22EA8643C667" name="X793B22EA8643C667"></a></p>
<h4>9.10 <span class="Heading">Automorphic Loops</span></h4>
<h4>9.11 <span class="Heading">Automorphic Loops</span></h4>
<p>The library named <em>automorphic</em> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively, including abelian groups), and commutative automorphic loops <span class="SimpleMath">\(Q\)</span> of order 243 possessing a central subloop <span class="SimpleMath">\(S\)</span> of order 3 such that <span class="SimpleMath">\(Q/S\)</span> is not the elementary abelian group of order 81 (there are 118451 such loops).</p>
<p>The library named <em>automorphic</em> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15) and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops).</p>
<p>It turns out that commutative automorphic loops of order 3, 9, 27 and 81 (but not 243) are in one-to-on correspondence with left Bruck loops of the respective orders, see <a href="chapBib_mj.html#biBGreer">[Gre14]</a>, <a href="chapBib_mj.html#biBStuhlVojtechovsky">[SV17]</a>. Only the left Bruck loops are stored in the library.</p>
<p><a id="X784FFA9E7FDA9F43" name="X784FFA9E7FDA9F43"></a></p>
<h5>9.10-1 AutomorphicLoop</h5>
<h5>9.11-1 AutomorphicLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AutomorphicLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th automorphic loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X843BD73F788049F7" name="X843BD73F788049F7"></a></p>
<h4>9.11 <span class="Heading">Interesting Loops</span></h4>
<h4>9.12 <span class="Heading">Interesting Loops</span></h4>
<p>The library named <em>interesting</em> contains some loops that are illustrative in the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a non-Moufang left Bol loop of order 16, the loop of sedenions of order 32 (sedenions generalize octonions), and the unique nonassociative simple right Bol loop of order 96 and exponent 2.</p>
<p><a id="X87F24AD3811910D3" name="X87F24AD3811910D3"></a></p>
<h5>9.11-1 InterestingLoop</h5>
<h5>9.12-1 InterestingLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; InterestingLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th interesting loop of order <var class="Arg">n</var> in the library.</p>
<p><a id="X864839227D5C0A90" name="X864839227D5C0A90"></a></p>
<h4>9.12 <span class="Heading">Libraries of Loops Up To Isotopism</span></h4>
<h4>9.13 <span class="Heading">Libraries of Loops Up To Isotopism</span></h4>
<p>For the library named <em>small</em> we also provide the corresponding library of loops up to isotopism. In general, given a library named <em>libname</em>, the corresponding library of loops up to isotopism is named <em>itp lib</em>, and the loops can be retrieved by the template <code class="code">ItpLibLoop(n,m)</code>.</p>
<p><a id="X850C4C01817A098D" name="X850C4C01817A098D"></a></p>
<h5>9.12-1 ItpSmallLoop</h5>
<h5>9.13-1 ItpSmallLoop</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ItpSmallLoop</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: The <var class="Arg">m</var>th small loop of order <var class="Arg">n</var> up to isotopism in the library.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 ); </span>
&lt;small loop 6/14&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 ); </span>
&lt;small loop 6/42&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 ); </span>
&lt;small loop 6/42&gt;
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SmallLoop( 6, 14 );</span>
&lt;small loop 6/14&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ItpSmallLoop( 6, 14 );</span>
&lt;small loop 6/42&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">LibraryLoop( "itp small", 6, 14 );</span>
&lt;small loop 6/42&gt;
</pre></div>
<p>Note that loops up to isotopism form a subset of the corresponding library of loops up to isomorphism. For instance, the above example shows that the 14th small loop of order 6 up to isotopism is in fact the 42nd small loop of order 6 up to isomorphism.</p>

7
doc/chapB.txt
View File

@ -105,9 +105,6 @@
( IsLeftAutomorphicLoop, IsAutomorphicLoop )
( IsRightAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsAutomorphicLoop )
( IsMiddleAutomorphicLoop, IsCommutative )
( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )
( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )
( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and
HasAntiautomorphicInverseProperty )
( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and
@ -120,9 +117,13 @@
( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )
( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )
( IsMiddleAutomorphicLoop, IsCommutative )
( IsLeftAutomorphicLoop, IsLeftBruckLoop )
( IsLeftAutomorphicLoop, IsLCCLoop )
( IsRightAutomorphicLoop, IsRightBruckLoop )
( IsRightAutomorphicLoop, IsRCCLoop )
( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )
( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )
( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )
( IsAutomorphicLoop, IsAssociative )

2
doc/chapB_mj.html
View File

File diff suppressed because one or more lines are too long

7
doc/chapBib.txt
View File

@ -42,6 +42,9 @@
less than 64, Nova Science Publishers Inc., Commack, NY (1999), xviii+287
pages.
[Gre14] Greer, M., A class of loops categorically isomorphic to Bruck loops
of odd order, Comm. Algebra, 42, 8 (2014), 3682–3697.
[GKN14] Grishkov, A., Kinyon, M. and Nagy, G. P., Solvability of commutative
automorphic loops, Proc. Amer. Math. Soc., 142, 9 (2014), 3029–3037.
@ -89,6 +92,10 @@
[SZ12] Slattery, M. and Zenisek, A., Moufang loops of order 243,
Commentationes Mathematicae Universitatis Carolinae, 53, 3 (2012), 423–428.
[SV17] Stuhl, I. and Vojtěchovský, P., Involutory latin quandles, Bruck
loops and commutative automorphic loops of odd prime power order,  (2017),
((preprint)).
[Voj06] Vojtěchovský, P., Toward the classification of Moufang loops of
order 64, European J. Combin., 27, 3 (2006), 444–460.

23
doc/chapBib_mj.html
View File

@ -164,6 +164,18 @@
</p>
<p><a id="biBGreer" name="biBGreer"></a></p>
<p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=3196069">Gre14</a></span>] <b class='BibAuthor'>Greer, M.</b>,
<i class='BibTitle'>A class of loops categorically isomorphic to Bruck loops of
odd order</i>,
<span class='BibJournal'>Comm. Algebra</span>,
<em class='BibVolume'>42</em> (<span class='BibNumber'>8</span>)
(<span class='BibYear'>2014</span>),
<span class='BibPages'>3682–3697</span>.
</p>
<p><a id="biBGrKiNa" name="biBGrKiNa"></a></p>
<p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=3223359">GKN14</a></span>] <b class='BibAuthor'>Grishkov, A., Kinyon, M. and Nagy, G. P.</b>,
@ -327,6 +339,17 @@
</p>
<p><a id="biBStuhlVojtechovsky" name="biBStuhlVojtechovsky"></a></p>
<p class='BibEntry'>
[<span class='BibKey'>SV17</span>] <b class='BibAuthor'>Stuhl, I. and Vojtěchovský, P.</b>,
<i class='BibTitle'>Involutory latin quandles, Bruck loops and commutative automorphic
loops of odd prime power order</i>,
<span class='BibJournal'></span>
(<span class='BibYear'>2017</span>)<br />
(<span class='BibNote'>preprint</span>).
</p>
<p><a id="biBVo" name="biBVo"></a></p>
<p class='BibEntry'>
[<span class='BibKeyLink'><a href="http://www.ams.org/mathscinet-getitem?mr=2206479">Voj06</a></span>] <b class='BibAuthor'>Vojtěchovský, P.</b>,

54
doc/chapInd.txt
View File

@ -11,7 +11,7 @@
alternative loop, left 7.4
alternative loop, right 7.4
antiautomorphic inverse property 7.2-5
AreEqualDiscriminators 6.11-9
AreEqualDiscriminators 6.11-11
AssociatedLeftBruckLoop 8.1-1
AssociatedRightBruckLoop 8.1-1
associator 2.5
@ -23,7 +23,7 @@
automorphic loop, left 7.7
automorphic loop, middle 7.7
automorphic loop, right 7.7
AutomorphicLoop 9.10-1
AutomorphicLoop 9.11-1
AutomorphismGroup 6.11-5
Bol loop, left 3.3
Bol loop, left 7.4
@ -39,7 +39,7 @@
Cayley table, canonical 4.3-1
CayleyTable 5.1-2
CayleyTableByPerms 4.6-1
CCLoop 9.6-3
CCLoop 9.7-3
center 2.3
Center 6.6-4
central series, lower 6.9-5
@ -47,7 +47,7 @@
Chein loop 8.2-3
cocycle 4.8
code loop 7.8-1
CodeLoop 9.4-1
CodeLoop 9.5-1
commutant 2.3
Commutant 6.6-3
commutator 2.5
@ -55,7 +55,7 @@
conjugacy closed loop 7.6
conjugacy closed loop, left 7.6
conjugacy closed loop, right 7.6
ConjugacyClosedLoop 9.6-3
ConjugacyClosedLoop 9.7-3
conjugation 6.5
coset 6.2-6
derived series 2.4
@ -64,7 +64,7 @@
DerivedSubloop 6.10-2
diassociative quasigroup 7.1-4
DirectProduct 4.11-1
Discriminator 6.11-8
Discriminator 6.11-10
DisplayLibraryInfo 9.1-3
distributive quasigroup 7.3-6
distributive quasigroup, left 7.3-6
@ -111,7 +111,7 @@
inner mapping group, middle 6.5
inner mapping group, right 2.2
InnerMappingGroup 6.5-3
InterestingLoop 9.11-1
InterestingLoop 9.12-1
IntoGroup 4.10-4
IntoLoop 4.10-3
IntoQuasigroup 4.10-1
@ -169,8 +169,8 @@
IsNilpotent 6.9-1
IsNormal 6.7-1
IsNuclearSquareLoop 7.4-11
IsomorphicCopyByNormalSubloop 6.11-7
IsomorphicCopyByPerm 6.11-6
IsomorphicCopyByNormalSubloop 6.11-9
IsomorphicCopyByPerm 6.11-8
isomorphism 2.6
IsomorphismLoops 6.11-2
IsomorphismQuasigroups 6.11-1
@ -206,16 +206,17 @@
IsSubquasigroup 6.2-3
IsTotallySymmetric 7.3-2
IsUnipotent 7.3-5
ItpSmallLoop 9.12-1
ItpSmallLoop 9.13-1
K loop, left 7.8-3
K loop, right 7.8-4
latin square 2.1
latin square 4.1
latin square, random 4.9
LC loop 7.4
LCCLoop 9.6-2
LCCLoop 9.7-2
LeftBolLoop 9.2-1
LeftConjugacyClosedLoop 9.6-2
LeftBruckLoop 9.3-1
LeftConjugacyClosedLoop 9.7-2
LeftDivision 5.2-1
LeftDivision 5.2-1
LeftDivision 5.2-1
@ -234,7 +235,7 @@
loop, LC 7.4
loop, Moufang 7.4
loop, Osborn 7.6-4
loop, Paige 9.8
loop, Paige 9.9
loop, RC 7.4
loop, Steiner 7.8-2
loop, alternative 7.4
@ -259,7 +260,7 @@
loop, nilpotent 2.4
loop, nilpotent 4.9-2
loop, nuclear square 7.4
loop, octonion 9.3-1
loop, octonion 9.4-1
loop, of Bol-Moufang type 7.4
loop, power alternative 7.5
loop, power associative 5.1-5
@ -271,7 +272,7 @@
loop, right conjugacy closed 7.6
loop, right nuclear square 7.4
loop, right power alternative 7.5
loop, sedenion 9.11
loop, sedenion 9.12
loop, simple 3.3
loop, simple 6.7-3
loop, solvable 2.4
@ -286,6 +287,7 @@
LoopByRightFolder 4.7-1
LoopByRightSection 4.6-3
LoopFromFile 4.5-1
LoopIsomorph 6.11-7
LoopMG2 8.2-3
LoopsUpToIsomorphism 6.11-4
LoopsUpToIsotopism 6.12-2
@ -299,7 +301,7 @@
modification, cyclic 8.2-1
modification, dihedral 8.2-2
Moufang loop 7.4
MoufangLoop 9.3-1
MoufangLoop 9.4-1
multiplication group 2.2
multiplication group, left 2.2
multiplication group, relative 6.4-2
@ -315,7 +317,7 @@
NilpotencyClassOfLoop 6.9-2
nilpotent loop 2.4
nilpotent loop, strongly 6.9-3
NilpotentLoop 9.9-1
NilpotentLoop 9.10-1
normal closure 6.7-2
normal subloop 6.7-1
NormalClosure 6.7-2
@ -332,7 +334,7 @@
nucleus, right 2.3
NucleusOfLoop 6.6-2
NucleusOfQuasigroup 6.6-2
octonion loop 9.3-1
octonion loop 9.4-1
One 5.1-3
OneLoopTableInGroup 8.4-3
OneLoopWithMltGroup 8.4-6
@ -342,8 +344,8 @@
OppositeLoop 4.12-1
OppositeQuasigroup 4.12-1
Osborn loop 7.6-4
Paige loop 9.8
PaigeLoop 9.8-1
Paige loop 9.9
PaigeLoop 9.9-1
Parent 6.1-1
PosInParent 6.1-3
Position 6.1-2
@ -373,18 +375,20 @@
QuasigroupByRightFolder 4.7-1
QuasigroupByRightSection 4.6-3
QuasigroupFromFile 4.5-1
QuasigroupIsomorph 6.11-6
QuasigroupsUpToIsomorphism 6.11-3
RandomLoop 4.9-1
RandomNilpotentLoop 4.9-2
RandomQuasigroup 4.9-1
RC loop 7.4
RCCLoop 9.6-1
RCCLoop 9.7-1
RelativeLeftMultiplicationGroup 6.4-2
RelativeMultiplicationGroup 6.4-2
RelativeRightMultiplicationGroup 6.4-2
RightBolLoop 9.2-2
RightBolLoopByExactGroupFactorization 8.1-3
RightConjugacyClosedLoop 9.6-1
RightBruckLoop 9.3-2
RightConjugacyClosedLoop 9.7-1
RightCosets 6.2-6
RightDivision 5.2-1
RightDivision 5.2-1
@ -400,7 +404,7 @@
RightTransversal 6.2-7
section, left 2.2
section, right 2.2
sedenion loop 9.11
sedenion loop 9.12
semisymmetric quasigroup 7.3-1
SetLoopElmName 3.4-1
SetQuasigroupElmName 3.4-1
@ -408,12 +412,12 @@
simple loop 6.7-3
Size 5.1-4
SmallGeneratingSet 5.5-3
SmallLoop 9.7-1
SmallLoop 9.8-1
solvability class 2.4
solvable loop 2.4
Steiner loop 7.8-2
Steiner quasigroup 7.3-4
SteinerLoop 9.5-1
SteinerLoop 9.6-1
strongly nilpotent loop 6.9-3
subloop 2.3
Subloop 6.2-2

54
doc/chapInd_mj.html
View File

@ -37,7 +37,7 @@ alternative loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
alternative loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
alternative loop, right <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
antiautomorphic inverse property <a href="chap7_mj.html#X8538D4638232DB51">7.2-5</a><br />
<code class="func">AreEqualDiscriminators</code> <a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9</a><br />
<code class="func">AreEqualDiscriminators</code> <a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11</a><br />
<code class="func">AssociatedLeftBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br />
<code class="func">AssociatedRightBruckLoop</code> <a href="chap8_mj.html#X8664CA927DD73DBE">8.1-1</a><br />
associator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
@ -49,7 +49,7 @@ automorphic loop <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, left <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, middle <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
automorphic loop, right <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
<code class="func">AutomorphicLoop</code> <a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1</a><br />
<code class="func">AutomorphicLoop</code> <a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1</a><br />
<code class="func">AutomorphismGroup</code> <a href="chap6_mj.html#X87677B0787B4461A">6.11-5</a><br />
Bol loop, left <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
Bol loop, left <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
@ -65,7 +65,7 @@ Cayley table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
Cayley table, canonical <a href="chap4_mj.html#X7971CCB87DAFF7B9">4.3-1</a><br />
<code class="func">CayleyTable</code> <a href="chap5_mj.html#X85457FA27DE7114D">5.1-2</a><br />
<code class="func">CayleyTableByPerms</code> <a href="chap4_mj.html#X7F94C8DD7E1A3470">4.6-1</a><br />
<code class="func">CCLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.6-3</a><br />
<code class="func">CCLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.7-3</a><br />
center <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Center</code> <a href="chap6_mj.html#X7C1FBE7A84DD4873">6.6-4</a><br />
central series, lower <a href="chap6_mj.html#X817BDBC2812992ED">6.9-5</a><br />
@ -73,7 +73,7 @@ central series, upper <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
Chein loop <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
cocycle <a href="chap4_mj.html#X8759431780AC81A9">4.8</a><br />
code loop <a href="chap7_mj.html#X790FA1188087D5C1">7.8-1</a><br />
<code class="func">CodeLoop</code> <a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1</a><br />
<code class="func">CodeLoop</code> <a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1</a><br />
commutant <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Commutant</code> <a href="chap6_mj.html#X7C8428DE791F3CE1">6.6-3</a><br />
commutator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
@ -81,7 +81,7 @@ commutator <a href="chap2_mj.html#X7E0849977869E53D">2.5</a><br />
conjugacy closed loop <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
conjugacy closed loop, left <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
conjugacy closed loop, right <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
<code class="func">ConjugacyClosedLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.6-3</a><br />
<code class="func">ConjugacyClosedLoop</code> <a href="chap9_mj.html#X798BC601843E8916">9.7-3</a><br />
conjugation <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br />
coset <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br />
derived series <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
@ -90,7 +90,7 @@ derived subloop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
<code class="func">DerivedSubloop</code> <a href="chap6_mj.html#X7A82DC4680DAD67C">6.10-2</a><br />
diassociative quasigroup <a href="chap7_mj.html#X872DCA027E1A4A1D">7.1-4</a><br />
<code class="func">DirectProduct</code> <a href="chap4_mj.html#X861BA02C7902A4F4">4.11-1</a><br />
<code class="func">Discriminator</code> <a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8</a><br />
<code class="func">Discriminator</code> <a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10</a><br />
<code class="func">DisplayLibraryInfo</code> <a href="chap9_mj.html#X7A64372E81E713B4">9.1-3</a><br />
distributive quasigroup <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br />
distributive quasigroup, left <a href="chap7_mj.html#X7B76FD6E878ED4F1">7.3-6</a><br />
@ -137,7 +137,7 @@ inner mapping group, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br
inner mapping group, middle <a href="chap6_mj.html#X8740D61178ACD217">6.5</a><br />
inner mapping group, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
<code class="func">InnerMappingGroup</code> <a href="chap6_mj.html#X82513A3B7C3A6420">6.5-3</a><br />
<code class="func">InterestingLoop</code> <a href="chap9_mj.html#X87F24AD3811910D3">9.11-1</a><br />
<code class="func">InterestingLoop</code> <a href="chap9_mj.html#X87F24AD3811910D3">9.12-1</a><br />
<code class="func">IntoGroup</code> <a href="chap4_mj.html#X7B5C6C64831B866E">4.10-4</a><br />
<code class="func">IntoLoop</code> <a href="chap4_mj.html#X7A59C36683118E5A">4.10-3</a><br />
<code class="func">IntoQuasigroup</code> <a href="chap4_mj.html#X84575A4B78CC545E">4.10-1</a><br />
@ -195,8 +195,8 @@ IsLoopElement <a href="chap3_mj.html#X86F02BBD87FEA1C6">3.1</a><br />
<code class="func">IsNilpotent</code> <a href="chap6_mj.html#X78A4B93781C96AAE">6.9-1</a><br />
<code class="func">IsNormal</code> <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br />
<code class="func">IsNuclearSquareLoop</code> <a href="chap7_mj.html#X796650088213229B">7.4-11</a><br />
<code class="func">IsomorphicCopyByNormalSubloop</code> <a href="chap6_mj.html#X8121DE3A78795040">6.11-7</a><br />
<code class="func">IsomorphicCopyByPerm</code> <a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6</a><br />
<code class="func">IsomorphicCopyByNormalSubloop</code> <a href="chap6_mj.html#X8121DE3A78795040">6.11-9</a><br />
<code class="func">IsomorphicCopyByPerm</code> <a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8</a><br />
isomorphism <a href="chap2_mj.html#X791066ED7DD9F254">2.6</a><br />
<code class="func">IsomorphismLoops</code> <a href="chap6_mj.html#X7D7B10D6836FCA9F">6.11-2</a><br />
<code class="func">IsomorphismQuasigroups</code> <a href="chap6_mj.html#X801067F67E5292F7">6.11-1</a><br />
@ -232,16 +232,17 @@ IsQuasigroupElement <a href="chap3_mj.html#X86F02BBD87FEA1C6">3.1</a><br />
<code class="func">IsSubquasigroup</code> <a href="chap6_mj.html#X87AC8B7E80CE9260">6.2-3</a><br />
<code class="func">IsTotallySymmetric</code> <a href="chap7_mj.html#X834F809B8060B754">7.3-2</a><br />
<code class="func">IsUnipotent</code> <a href="chap7_mj.html#X7CA3DCA07B6CB9BD">7.3-5</a><br />
<code class="func">ItpSmallLoop</code> <a href="chap9_mj.html#X850C4C01817A098D">9.12-1</a><br />
<code class="func">ItpSmallLoop</code> <a href="chap9_mj.html#X850C4C01817A098D">9.13-1</a><br />
K loop, left <a href="chap7_mj.html#X85F1BD4280E44F5B">7.8-3</a><br />
K loop, right <a href="chap7_mj.html#X857B373E7B4E0519">7.8-4</a><br />
latin square <a href="chap2_mj.html#X80243DE5826583B8">2.1</a><br />
latin square <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
latin square, random <a href="chap4_mj.html#X7AE29A1A7AA5C25A">4.9</a><br />
LC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">LCCLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2</a><br />
<code class="func">LCCLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2</a><br />
<code class="func">LeftBolLoop</code> <a href="chap9_mj.html#X7EE99F647C537994">9.2-1</a><br />
<code class="func">LeftConjugacyClosedLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.6-2</a><br />
<code class="func">LeftBruckLoop</code> <a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1</a><br />
<code class="func">LeftConjugacyClosedLoop</code> <a href="chap9_mj.html#X80AB8B107D55FB19">9.7-2</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">LeftDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
@ -260,7 +261,7 @@ loop, Chein <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
loop, LC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Moufang <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Osborn <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br />
loop, Paige <a href="chap9_mj.html#X8135C8FD8714C606">9.8</a><br />
loop, Paige <a href="chap9_mj.html#X8135C8FD8714C606">9.9</a><br />
loop, RC <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, Steiner <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br />
loop, alternative <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
@ -285,7 +286,7 @@ loop, middle nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><b
loop, nilpotent <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
loop, nilpotent <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br />
loop, nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, octonion <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br />
loop, octonion <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
loop, of Bol-Moufang type <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br />
loop, power associative <a href="chap5_mj.html#X7D44470C7DA59C1C">5.1-5</a><br />
@ -297,7 +298,7 @@ loop, right automorphic <a href="chap7_mj.html#X793B22EA8643C667">7.7</a><br />
loop, right conjugacy closed <a href="chap7_mj.html#X8176B2C47A4629CD">7.6</a><br />
loop, right nuclear square <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
loop, right power alternative <a href="chap7_mj.html#X83A501387E1AC371">7.5</a><br />
loop, sedenion <a href="chap9_mj.html#X843BD73F788049F7">9.11</a><br />
loop, sedenion <a href="chap9_mj.html#X843BD73F788049F7">9.12</a><br />
loop, simple <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
loop, simple <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br />
loop, solvable <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
@ -312,6 +313,7 @@ loop table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
<code class="func">LoopByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br />
<code class="func">LoopByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br />
<code class="func">LoopFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br />
<code class="func">LoopIsomorph</code> <a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7</a><br />
<code class="func">LoopMG2</code> <a href="chap8_mj.html#X7CC6CDB786E9BBA0">8.2-3</a><br />
<code class="func">LoopsUpToIsomorphism</code> <a href="chap6_mj.html#X8308F38283C61B20">6.11-4</a><br />
<code class="func">LoopsUpToIsotopism</code> <a href="chap6_mj.html#X841E540B7A7EF29F">6.12-2</a><br />
@ -325,7 +327,7 @@ modification, Moufang <a href="chap8_mj.html#X819F82737C2A860D">8.2</a><br />
modification, cyclic <a href="chap8_mj.html#X7B3165C083709831">8.2-1</a><br />
modification, dihedral <a href="chap8_mj.html#X7D7717C587BC2D1E">8.2-2</a><br />
Moufang loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">MoufangLoop</code> <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br />
<code class="func">MoufangLoop</code> <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
multiplication group <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
multiplication group, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
multiplication group, relative <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
@ -341,7 +343,7 @@ nilpotence class <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
<code class="func">NilpotencyClassOfLoop</code> <a href="chap6_mj.html#X7D5FC62581A99482">6.9-2</a><br />
nilpotent loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
nilpotent loop, strongly <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br />
<code class="func">NilpotentLoop</code> <a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1</a><br />
<code class="func">NilpotentLoop</code> <a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1</a><br />
normal closure <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br />
normal subloop <a href="chap6_mj.html#X838186F9836F678C">6.7-1</a><br />
<code class="func">NormalClosure</code> <a href="chap6_mj.html#X7BDEA0A98720D1BB">6.7-2</a><br />
@ -358,7 +360,7 @@ nucleus, middle <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
nucleus, right <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">NucleusOfLoop</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br />
<code class="func">NucleusOfQuasigroup</code> <a href="chap6_mj.html#X84D389677A91C290">6.6-2</a><br />
octonion loop <a href="chap9_mj.html#X81E82098822543EE">9.3-1</a><br />
octonion loop <a href="chap9_mj.html#X81E82098822543EE">9.4-1</a><br />
<code class="func">One</code> <a href="chap5_mj.html#X8129A6877FFD804B">5.1-3</a><br />
<code class="func">OneLoopTableInGroup</code> <a href="chap8_mj.html#X7BFFC66A824BA6AA">8.4-3</a><br />
<code class="func">OneLoopWithMltGroup</code> <a href="chap8_mj.html#X8266DE05824226E6">8.4-6</a><br />
@ -368,8 +370,8 @@ opposite quasigroup <a href="chap4_mj.html#X7865FC8D7854C2E3">4.12</a><br />
<code class="func">OppositeLoop</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br />
<code class="func">OppositeQuasigroup</code> <a href="chap4_mj.html#X87B6AED47EE2BCD3">4.12-1</a><br />
Osborn loop <a href="chap7_mj.html#X8655956878205FC1">7.6-4</a><br />
Paige loop <a href="chap9_mj.html#X8135C8FD8714C606">9.8</a><br />
<code class="func">PaigeLoop</code> <a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1</a><br />
Paige loop <a href="chap9_mj.html#X8135C8FD8714C606">9.9</a><br />
<code class="func">PaigeLoop</code> <a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1</a><br />
<code class="func">Parent</code> <a href="chap6_mj.html#X7BC856CC7F116BB0">6.1-1</a><br />
<code class="func">PosInParent</code> <a href="chap6_mj.html#X832295DE866E44EE">6.1-3</a><br />
<code class="func">Position</code> <a href="chap6_mj.html#X79975EC6783B4293">6.1-2</a><br />
@ -399,18 +401,20 @@ quasigroup table <a href="chap4_mj.html#X7DE8405B82BC36A9">4.1</a><br />
<code class="func">QuasigroupByRightFolder</code> <a href="chap4_mj.html#X83168E62861F70AB">4.7-1</a><br />
<code class="func">QuasigroupByRightSection</code> <a href="chap4_mj.html#X80B436ED7CC0749E">4.6-3</a><br />
<code class="func">QuasigroupFromFile</code> <a href="chap4_mj.html#X81A1DB918057933E">4.5-1</a><br />
<code class="func">QuasigroupIsomorph</code> <a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6</a><br />
<code class="func">QuasigroupsUpToIsomorphism</code> <a href="chap6_mj.html#X82373C5479574F22">6.11-3</a><br />
<code class="func">RandomLoop</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br />
<code class="func">RandomNilpotentLoop</code> <a href="chap4_mj.html#X817132C887D3FD3A">4.9-2</a><br />
<code class="func">RandomQuasigroup</code> <a href="chap4_mj.html#X8271C0F5786B6FA9">4.9-1</a><br />
RC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">RCCLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.6-1</a><br />
<code class="func">RCCLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.7-1</a><br />
<code class="func">RelativeLeftMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RelativeMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RelativeRightMultiplicationGroup</code> <a href="chap6_mj.html#X847256B779E1E7E5">6.4-2</a><br />
<code class="func">RightBolLoop</code> <a href="chap9_mj.html#X8774304282654C58">9.2-2</a><br />
<code class="func">RightBolLoopByExactGroupFactorization</code> <a href="chap8_mj.html#X7DCA64807F899127">8.1-3</a><br />
<code class="func">RightConjugacyClosedLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.6-1</a><br />
<code class="func">RightBruckLoop</code> <a href="chap9_mj.html#X798DD7CF871F648F">9.3-2</a><br />
<code class="func">RightConjugacyClosedLoop</code> <a href="chap9_mj.html#X806B2DE67990E42F">9.7-1</a><br />
<code class="func">RightCosets</code> <a href="chap6_mj.html#X835F48248571364F">6.2-6</a><br />
<code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
<code class="func">RightDivision</code> <a href="chap5_mj.html#X7D5956967BCC1834">5.2-1</a><br />
@ -426,7 +430,7 @@ RC loop <a href="chap7_mj.html#X780D907986EBA6C7">7.4</a><br />
<code class="func">RightTransversal</code> <a href="chap6_mj.html#X85C65D06822E716F">6.2-7</a><br />
section, left <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
section, right <a href="chap2_mj.html#X7EC01B437CC2B2C9">2.2</a><br />
sedenion loop <a href="chap9_mj.html#X843BD73F788049F7">9.11</a><br />
sedenion loop <a href="chap9_mj.html#X843BD73F788049F7">9.12</a><br />
semisymmetric quasigroup <a href="chap7_mj.html#X834848ED85F9012B">7.3-1</a><br />
<code class="func">SetLoopElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br />
<code class="func">SetQuasigroupElmName</code> <a href="chap3_mj.html#X7A7EB1B579273D07">3.4-1</a><br />
@ -434,12 +438,12 @@ simple loop <a href="chap3_mj.html#X87E49ED884FA6DC4">3.3</a><br />
simple loop <a href="chap6_mj.html#X7D8E63A7824037CC">6.7-3</a><br />
<code class="func">Size</code> <a href="chap5_mj.html#X858ADA3B7A684421">5.1-4</a><br />
<code class="func">SmallGeneratingSet</code> <a href="chap5_mj.html#X814DBABC878D5232">5.5-3</a><br />
<code class="func">SmallLoop</code> <a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1</a><br />
<code class="func">SmallLoop</code> <a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1</a><br />
solvability class <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
solvable loop <a href="chap2_mj.html#X869CBCE381E2C422">2.4</a><br />
Steiner loop <a href="chap7_mj.html#X793600C9801F4F62">7.8-2</a><br />
Steiner quasigroup <a href="chap7_mj.html#X83DE7DD77C056C1F">7.3-4</a><br />
<code class="func">SteinerLoop</code> <a href="chap9_mj.html#X87C235457E859AF4">9.5-1</a><br />
<code class="func">SteinerLoop</code> <a href="chap9_mj.html#X87C235457E859AF4">9.6-1</a><br />
strongly nilpotent loop <a href="chap6_mj.html#X7E7C2D117B55F6A0">6.9-3</a><br />
subloop <a href="chap2_mj.html#X83EDF04F7952143F">2.3</a><br />
<code class="func">Subloop</code> <a href="chap6_mj.html#X84E6744E804AE830">6.2-2</a><br />

161
doc/loops.bbl
View File

@ -1,161 +0,0 @@
\begin{thebibliography}{DBGV12}
\bibitem[Art59]{Ar}
R.~Artzy.
\newblock On automorphic-inverse properties in loops.
\newblock {\em Proc. Amer. Math. Soc.}, 10:588{\textendash}591, 1959.
\bibitem[Art15]{Artic}
K.~Artic.
\newblock {\em On conjugacy closed loops and conjugacy closed loop folders}.
\newblock PhD thesis, RWTH Aachen University, 2015.
\bibitem[BP56]{BrPa}
R.~H. Bruck and L.~J. Paige.
\newblock Loops whose inner mappings are automorphisms.
\newblock {\em Ann. of Math. (2)}, 63:308{\textendash}323, 1956.
\bibitem[Bru58]{Br}
R.~H. Bruck.
\newblock {\em A survey of binary systems}.
\newblock Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft
20. Reihe: Gruppentheorie. Springer Verlag, Berlin, 1958.
\bibitem[CD05]{CsDr}
P.~Cs{\"o}rg{\H o} and A.~Dr{\a'a}pal.
\newblock Left conjugacy closed loops of nilpotency class two.
\newblock {\em Results Math.}, 47(3-4):242{\textendash}265, 2005.
\bibitem[CR99]{CoRo}
C.~J. Colbourn and A.~Rosa.
\newblock {\em Triple systems}.
\newblock Oxford Mathematical Monographs. The Clarendon Press Oxford University
Press, New York, 1999.
\bibitem[DBGV12]{BaGrVo}
D.~A.~S. De~Barros, A.~Grishkov, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Commutative automorphic loops of order {$p^3$}.
\newblock {\em J. Algebra Appl.}, 11(5):1250100, 15, 2012.
\bibitem[Dr{\a'a}03]{DrapalCD}
A.~Dr{\a'a}pal.
\newblock Cyclic and dihedral constructions of even order.
\newblock {\em Comment. Math. Univ. Carolin.}, 44(4):593{\textendash}614, 2003.
\bibitem[DV06]{DrVo}
A.~Dr{\a'a}pal and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Moufang loops that share associator and three quarters of their
multiplication tables.
\newblock {\em Rocky Mountain J. Math.}, 36(2):425{\textendash}455, 2006.
\bibitem[DV09]{DaVo}
D.~Daly and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Enumeration of nilpotent loops via cohomology.
\newblock {\em J. Algebra}, 322(11):4080{\textendash}4098, 2009.
\bibitem[Fen69]{Fe}
F.~Fenyves.
\newblock Extra loops. {II}. {O}n loops with identities of {B}ol-{M}oufang
type.
\newblock {\em Publ. Math. Debrecen}, 16:187{\textendash}192, 1969.
\bibitem[GKN14]{GrKiNa}
A.~Grishkov, M.~Kinyon, and G.~P. Nagy.
\newblock Solvability of commutative automorphic loops.
\newblock {\em Proc. Amer. Math. Soc.}, 142(9):3029{\textendash}3037, 2014.
\bibitem[GMR99]{Go}
E.~G. Goodaire, S.~May, and M.~Raman.
\newblock {\em The {M}oufang loops of order less than 64}.
\newblock Nova Science Publishers Inc., Commack, NY, 1999.
\bibitem[JKNV11]{JoKiNaVo}
K.~W. Johnson, M.~K. Kinyon, G.~P. Nagy, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Searching for small simple automorphic loops.
\newblock {\em LMS J. Comput. Math.}, 14:200{\textendash}213, 2011.
\bibitem[JKV12]{JeKiVo}
P.~Jedli{\v c}ka, M.~Kinyon, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Nilpotency in automorphic loops of prime power order.
\newblock {\em J. Algebra}, 350:64{\textendash}76, 2012.
\bibitem[JM96]{JaMa}
M.~T. Jacobson and P.~Matthews.
\newblock Generating uniformly distributed random {L}atin squares.
\newblock {\em J. Combin. Des.}, 4(6):405{\textendash}437, 1996.
\bibitem[KKP02]{KiKuPh}
M.~K. Kinyon, K.~Kunen, and J.~D. Phillips.
\newblock Every diassociative {$A$}-loop is {M}oufang.
\newblock {\em Proc. Amer. Math. Soc.}, 130(3):619{\textendash}624, 2002.
\bibitem[KKPV16]{KiKuPhVo}
M.~K. Kinyon, K.~Kunen, J.~D. Phillips, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The structure of automorphic loops.
\newblock {\em Trans. Amer. Math. Soc.}, 368(12):8901{\textendash}8927, 2016.
\bibitem[KNV15]{KiNaVo2015}
M.~K. Kinyon, G.~P. Nagy, and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Bol loops and bruck loops of order $pq$.
\newblock 2015.
\newblock preprint.
\bibitem[Kun00]{Kun}
K.~Kunen.
\newblock The structure of conjugacy closed loops.
\newblock {\em Trans. Amer. Math. Soc.}, 352(6):2889{\textendash}2911, 2000.
\bibitem[Lie87]{Li}
M.~W. Liebeck.
\newblock The classification of finite simple {M}oufang loops.
\newblock {\em Math. Proc. Cambridge Philos. Soc.}, 102(1):33{\textendash}47,
1987.
\bibitem[Moo]{Mo}
G.~E. Moorhouse.
\newblock Bol loops of small order.
\newblock http://www.uwyo.edu/moorhouse/pub/bol/.
\bibitem[NV03]{NaVo2003}
G.~P. Nagy and P.~Vojt{\v e}chovsk{\a'y}.
\newblock Octonions, simple {M}oufang loops and triality.
\newblock {\em Quasigroups Related Systems}, 10:65{\textendash}94, 2003.
\bibitem[NV07]{NaVo2007}
G.~P. Nagy and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The {M}oufang loops of order 64 and 81.
\newblock {\em J. Symbolic Comput.}, 42(9):871{\textendash}883, 2007.
\bibitem[Pfl90]{Pf}
H.~O. Pflugfelder.
\newblock {\em Quasigroups and loops: introduction}, volume~7 of {\em Sigma
Series in Pure Mathematics}.
\newblock Heldermann Verlag, Berlin, 1990.
\bibitem[PV05]{PhiVoj}
J.~D. Phillips and P.~Vojt{\v e}chovsk{\a'y}.
\newblock The varieties of loops of {B}ol-{M}oufang type.
\newblock {\em Algebra Universalis}, 54(3):259{\textendash}271, 2005.
\bibitem[SZ12]{SlZe2011}
M.~Slattery and A.~Zenisek.
\newblock Moufang loops of order 243.
\newblock {\em Commentationes Mathematicae Universitatis Carolinae},
53(3):423{\textendash}428, 2012.
\bibitem[Voj06]{Vo}
P.~Vojt{\v e}chovsk{\a'y}.
\newblock Toward the classification of {M}oufang loops of order 64.
\newblock {\em European J. Combin.}, 27(3):444{\textendash}460, 2006.
\bibitem[Voj15]{VoQRS}
P.~Vojt{\v e}chovsk{\a'y}.
\newblock Three lectures on automorphic loops.
\newblock {\em Quasigroups Related Systems}, 23(1):129{\textendash}163, 2015.
\bibitem[WJ75]{Wi}
R.~L. Wilson~Jr.
\newblock Quasidirect products of quasigroups.
\newblock {\em Comm. Algebra}, 3(9):835{\textendash}850, 1975.
\end{thebibliography}

34
doc/loops.bib
View File

@ -260,6 +260,27 @@ Publishers, 1999.
MRNUMBER = {1689624 (2000a:20147)},
}
\bibitem{Greer}
Mark Greer.
\newblock{\it A class of loops categorically isomorphic to Bruck loops of odd order},
Comm. Algebra {42} (2014), 3682--3697.
@article {Greer,
AUTHOR = {Greer, Mark},
TITLE = {A class of loops categorically isomorphic to {B}ruck loops of odd order},
JOURNAL = {Comm. Algebra},
FJOURNAL = {Communications in Algebra},
VOLUME = {42},
YEAR = {2014},
NUMBER = {8},
PAGES = {3682--3697},
ISSN = {0092-7872},
MRCLASS = {20N05},
MRNUMBER = {3196069},
MRREVIEWER = {Anil Kumar V.},
URL = {https://doi.org/10.1080/00927872.2013.791304},
}
\bibitem{GrKiNa}
Alexander Grishkov, Michael Kinyon and G\'abor Nagy.
\newblock {\it Solvability of commutative automorphic loops},
@ -591,6 +612,19 @@ preprint.
PAGES = {423--428},
}
\bibitem{StuhlVojtechovsky}
Izabella Stuhl and Petr Vojt\v{e}chovsk\'y.
\newblock {\it Involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order},
preprint, 2017.
@article {StuhlVojtechovsky,
AUTHOR = {Stuhl, Izabella and Vojt{\v{e}}chovsk{\'y}, Petr},
TITLE = {Involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order},
JOURNAL = {},
YEAR = {2017},
NOTE = {preprint},
}
\bibitem{Vo}
Petr Vojt\v{e}chovsk\'y.
\newblock {\it Toward the classification of Moufang loops of order $64$},

5
doc/loops.blg
View File

@ -1,5 +0,0 @@
This is BibTeX, Version 0.99dThe top-level auxiliary file: loops.aux
The style file: alpha.bst
Database file #1: loops_bib.xml.bib
Warning--empty journal in KiNaVo2015
(There was 1 warning)

35
doc/loops.brf
View File

@ -1,35 +0,0 @@
\backcite {Br}{{8}{2}{chapter.2}}
\backcite {Pf}{{8}{2}{chapter.2}}
\backcite {JaMa}{{19}{4.9}{section.4.9}}
\backcite {Vo}{{35}{6.11.8}{subsection.6.11.8}}
\backcite {Ar}{{37}{7.2.4}{subsection.7.2.4}}
\backcite {Fe}{{39}{7.4}{section.7.4}}
\backcite {PhiVoj}{{39}{7.4}{section.7.4}}
\backcite {PhiVoj}{{39}{7.4}{section.7.4}}
\backcite {BrPa}{{43}{7.7}{section.7.7}}
\backcite {BrPa}{{43}{7.7}{section.7.7}}
\backcite {JoKiNaVo}{{43}{7.7}{section.7.7}}
\backcite {KiKuPh}{{43}{7.7}{section.7.7}}
\backcite {KiKuPhVo}{{43}{7.7}{section.7.7}}
\backcite {GrKiNa}{{43}{7.7}{section.7.7}}
\backcite {VoQRS}{{43}{7.7}{section.7.7}}
\backcite {JeKiVo}{{43}{7.7}{section.7.7}}
\backcite {BaGrVo}{{43}{7.7}{section.7.7}}
\backcite {VoQRS}{{43}{7.7}{section.7.7}}
\backcite {DrapalCD}{{46}{8.2}{section.8.2}}
\backcite {DrVo}{{46}{8.2}{section.8.2}}
\backcite {NaVo2003}{{46}{8.3}{section.8.3}}
\backcite {Mo}{{50}{9.2}{section.9.2}}
\backcite {KiNaVo2015}{{50}{9.2}{section.9.2}}
\backcite {Go}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {NaVo2007}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {SlZe2011}{{50}{9.3.1}{subsection.9.3.1}}
\backcite {NaVo2007}{{51}{9.4}{section.9.4}}
\backcite {CoRo}{{51}{9.5}{section.9.5}}
\backcite {Artic}{{51}{9.6}{section.9.6}}
\backcite {Kun}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {CsDr}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Wi}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Kun}{{52}{9.6.2}{subsection.9.6.2}}
\backcite {Li}{{53}{9.8}{section.9.8}}
\backcite {DaVo}{{53}{9.9}{section.9.9}}

427
doc/loops.idx
View File

@ -1,427 +0,0 @@
\indexentry{groupoid|hyperpage}{8}
\indexentry{magma|hyperpage}{8}
\indexentry{neutral element|hyperpage}{8}
\indexentry{identity!element|hyperpage}{8}
\indexentry{inverse!two-sided|hyperpage}{8}
\indexentry{group|hyperpage}{8}
\indexentry{quasigroup|hyperpage}{8}
\indexentry{latin square|hyperpage}{8}
\indexentry{loop|hyperpage}{8}
\indexentry{translation!left|hyperpage}{8}
\indexentry{translation!right|hyperpage}{8}
\indexentry{division!left|hyperpage}{8}
\indexentry{division!right|hyperpage}{8}
\indexentry{section!left|hyperpage}{8}
\indexentry{section!right|hyperpage}{8}
\indexentry{multiplication group!left|hyperpage}{9}
\indexentry{multiplication group!right|hyperpage}{9}
\indexentry{multiplication group|hyperpage}{9}
\indexentry{inner mapping group!left|hyperpage}{9}
\indexentry{inner mapping group!right|hyperpage}{9}
\indexentry{inner mapping group|hyperpage}{9}
\indexentry{subquasigroup|hyperpage}{9}
\indexentry{subloop|hyperpage}{9}
\indexentry{nucleus!left|hyperpage}{9}
\indexentry{nucleus!middle|hyperpage}{9}
\indexentry{nucleus!right|hyperpage}{9}
\indexentry{nucleus|hyperpage}{9}
\indexentry{commutant|hyperpage}{9}
\indexentry{center|hyperpage}{9}
\indexentry{subloop!normal|hyperpage}{9}
\indexentry{nilpotence class|hyperpage}{9}
\indexentry{nilpotent loop|hyperpage}{9}
\indexentry{loop!nilpotent|hyperpage}{9}
\indexentry{central series!upper|hyperpage}{9}
\indexentry{derived subloop|hyperpage}{9}
\indexentry{solvability class|hyperpage}{9}
\indexentry{solvable loop|hyperpage}{9}
\indexentry{loop!solvable|hyperpage}{9}
\indexentry{derived series|hyperpage}{9}
\indexentry{commutator|hyperpage}{9}
\indexentry{associator|hyperpage}{9}
\indexentry{associator subloop|hyperpage}{9}
\indexentry{homomorphism|hyperpage}{9}
\indexentry{isomorphism|hyperpage}{9}
\indexentry{homotopism|hyperpage}{10}
\indexentry{isotopism|hyperpage}{10}
\indexentry{isotopism!principal|hyperpage}{10}
\indexentry{loop isotope!principal|hyperpage}{10}
\indexentry{IsQuasigroupElement|hyperpage}{11}
\indexentry{IsLoopElement|hyperpage}{11}
\indexentry{IsQuasigroup|hyperpage}{11}
\indexentry{IsLoop|hyperpage}{11}
\indexentry{Bol loop!left|hyperpage}{12}
\indexentry{loop!left Bol|hyperpage}{12}
\indexentry{simple loop|hyperpage}{12}
\indexentry{loop!simple|hyperpage}{12}
\indexentry{SetQuasigroupElmName@\texttt {SetQuasigroupElmName}|hyperpage}{13}
\indexentry{SetLoopElmName@\texttt {SetLoopElmName}|hyperpage}{13}
\indexentry{Cayley table|hyperpage}{14}
\indexentry{multiplication table|hyperpage}{14}
\indexentry{quasigroup table|hyperpage}{14}
\indexentry{latin square|hyperpage}{14}
\indexentry{loop table|hyperpage}{14}
\indexentry{IsQuasigroupTable@\texttt {IsQuasigroupTable}|hyperpage}{14}
\indexentry{IsQuasigroupCayleyTable@\texttt {IsQuasigroupCayleyTable}|hyperpage}{14}
\indexentry{IsLoopTable@\texttt {IsLoopTable}|hyperpage}{14}
\indexentry{IsLoopCayleyTable@\texttt {IsLoopCayleyTable}|hyperpage}{14}
\indexentry{CanonicalCayleyTable@\texttt {CanonicalCayleyTable}|hyperpage}{15}
\indexentry{Cayley table!canonical|hyperpage}{15}
\indexentry{CanonicalCopy@\texttt {CanonicalCopy}|hyperpage}{15}
\indexentry{NormalizedQuasigroupTable@\texttt {NormalizedQuasigroupTable}|hyperpage}{15}
\indexentry{QuasigroupByCayleyTable@\texttt {QuasigroupByCayleyTable}|hyperpage}{15}
\indexentry{LoopByCayleyTable@\texttt {LoopByCayleyTable}|hyperpage}{15}
\indexentry{QuasigroupFromFile@\texttt {QuasigroupFromFile}|hyperpage}{17}
\indexentry{LoopFromFile@\texttt {LoopFromFile}|hyperpage}{17}
\indexentry{CayleyTableByPerms@\texttt {CayleyTableByPerms}|hyperpage}{17}
\indexentry{QuasigroupByLeftSection@\texttt {QuasigroupByLeftSection}|hyperpage}{17}
\indexentry{LoopByLeftSection@\texttt {LoopByLeftSection}|hyperpage}{17}
\indexentry{QuasigroupByRightSection@\texttt {QuasigroupByRightSection}|hyperpage}{17}
\indexentry{LoopByRightSection@\texttt {LoopByRightSection}|hyperpage}{17}
\indexentry{folder!quasigroup|hyperpage}{18}
\indexentry{QuasigroupByRightFolder@\texttt {QuasigroupByRightFolder}|hyperpage}{18}
\indexentry{LoopByRightFolder@\texttt {LoopByRightFolder}|hyperpage}{18}
\indexentry{extension|hyperpage}{18}
\indexentry{extension!nuclear|hyperpage}{18}
\indexentry{cocycle|hyperpage}{18}
\indexentry{NuclearExtension@\texttt {NuclearExtension}|hyperpage}{18}
\indexentry{LoopByExtension@\texttt {LoopByExtension}|hyperpage}{18}
\indexentry{latin square!random|hyperpage}{19}
\indexentry{RandomQuasigroup@\texttt {RandomQuasigroup}|hyperpage}{19}
\indexentry{RandomLoop@\texttt {RandomLoop}|hyperpage}{19}
\indexentry{RandomNilpotentLoop@\texttt {RandomNilpotentLoop}|hyperpage}{19}
\indexentry{loop!nilpotent|hyperpage}{19}
\indexentry{IntoQuasigroup@\texttt {IntoQuasigroup}|hyperpage}{20}
\indexentry{PrincipalLoopIsotope@\texttt {PrincipalLoopIsotope}|hyperpage}{20}
\indexentry{IntoLoop@\texttt {IntoLoop}|hyperpage}{20}
\indexentry{IntoGroup@\texttt {IntoGroup}|hyperpage}{20}
\indexentry{DirectProduct@\texttt {DirectProduct}|hyperpage}{21}
\indexentry{opposite quasigroup|hyperpage}{21}
\indexentry{quasigroup!opposite|hyperpage}{21}
\indexentry{Opposite@\texttt {Opposite}|hyperpage}{21}
\indexentry{OppositeQuasigroup@\texttt {OppositeQuasigroup}|hyperpage}{21}
\indexentry{OppositeLoop@\texttt {OppositeLoop}|hyperpage}{21}
\indexentry{Elements@\texttt {Elements}|hyperpage}{22}
\indexentry{CayleyTable@\texttt {CayleyTable}|hyperpage}{22}
\indexentry{One@\texttt {One}|hyperpage}{22}
\indexentry{Size@\texttt {Size}|hyperpage}{22}
\indexentry{Exponent@\texttt {Exponent}|hyperpage}{23}
\indexentry{loop!power associative|hyperpage}{23}
\indexentry{power associative loop|hyperpage}{23}
\indexentry{exponent|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{LeftDivision@\texttt {LeftDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{RightDivision@\texttt {RightDivision}|hyperpage}{23}
\indexentry{LeftDivisionCayleyTable@\texttt {LeftDivisionCayleyTable}|hyperpage}{23}
\indexentry{RightDivisionCayleyTable@\texttt {RightDivisionCayleyTable}|hyperpage}{23}
\indexentry{inverse!left|hyperpage}{24}
\indexentry{inverse!right|hyperpage}{24}
\indexentry{inverse|hyperpage}{24}
\indexentry{LeftInverse@\texttt {LeftInverse}|hyperpage}{24}
\indexentry{RightInverse@\texttt {RightInverse}|hyperpage}{24}
\indexentry{Inverse@\texttt {Inverse}|hyperpage}{24}
\indexentry{Associator@\texttt {Associator}|hyperpage}{24}
\indexentry{Commutator@\texttt {Commutator}|hyperpage}{24}
\indexentry{GeneratorsOfQuasigroup@\texttt {GeneratorsOfQuasigroup}|hyperpage}{24}
\indexentry{GeneratorsOfLoop@\texttt {GeneratorsOfLoop}|hyperpage}{24}
\indexentry{GeneratorsSmallest@\texttt {GeneratorsSmallest}|hyperpage}{25}
\indexentry{SmallGeneratingSet@\texttt {SmallGeneratingSet}|hyperpage}{25}
\indexentry{Parent@\texttt {Parent}|hyperpage}{26}
\indexentry{Position@\texttt {Position}|hyperpage}{26}
\indexentry{PosInParent@\texttt {PosInParent}|hyperpage}{27}
\indexentry{Subquasigroup@\texttt {Subquasigroup}|hyperpage}{27}
\indexentry{Subloop@\texttt {Subloop}|hyperpage}{27}
\indexentry{IsSubquasigroup@\texttt {IsSubquasigroup}|hyperpage}{27}
\indexentry{IsSubloop@\texttt {IsSubloop}|hyperpage}{27}
\indexentry{AllSubquasigroups@\texttt {AllSubquasigroups}|hyperpage}{27}
\indexentry{AllSubloops@\texttt {AllSubloops}|hyperpage}{28}
\indexentry{RightCosets@\texttt {RightCosets}|hyperpage}{28}
\indexentry{coset|hyperpage}{28}
\indexentry{RightTransversal@\texttt {RightTransversal}|hyperpage}{28}
\indexentry{transversal|hyperpage}{28}
\indexentry{LeftTranslation@\texttt {LeftTranslation}|hyperpage}{28}
\indexentry{RightTranslation@\texttt {RightTranslation}|hyperpage}{28}
\indexentry{LeftSection@\texttt {LeftSection}|hyperpage}{28}
\indexentry{RightSection@\texttt {RightSection}|hyperpage}{28}
\indexentry{LeftMultiplicationGroup@\texttt {LeftMultiplicationGroup}|hyperpage}{29}
\indexentry{RightMultiplicationGroup@\texttt {RightMultiplicationGroup}|hyperpage}{29}
\indexentry{MultiplicationGroup@\texttt {MultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeLeftMultiplicationGroup@\texttt {RelativeLeftMultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeRightMultiplicationGroup@\texttt {RelativeRightMultiplicationGroup}|hyperpage}{29}
\indexentry{RelativeMultiplicationGroup@\texttt {RelativeMultiplicationGroup}|hyperpage}{29}
\indexentry{multiplication group!relative left|hyperpage}{29}
\indexentry{multiplication group!relative right |hyperpage}{29}
\indexentry{multiplication group!relative|hyperpage}{29}
\indexentry{inner mapping!left|hyperpage}{29}
\indexentry{inner mapping!right|hyperpage}{29}
\indexentry{conjugation|hyperpage}{30}
\indexentry{inner mapping!middle|hyperpage}{30}
\indexentry{inner mapping group!middle|hyperpage}{30}
\indexentry{LeftInnerMapping@\texttt {LeftInnerMapping}|hyperpage}{30}
\indexentry{RightInnerMapping@\texttt {RightInnerMapping}|hyperpage}{30}
\indexentry{MiddleInnerMapping@\texttt {MiddleInnerMapping}|hyperpage}{30}
\indexentry{LeftInnerMappingGroup@\texttt {LeftInnerMappingGroup}|hyperpage}{30}
\indexentry{RightInnerMappingGroup@\texttt {RightInnerMappingGroup}|hyperpage}{30}
\indexentry{MiddleInnerMappingGroup@\texttt {MiddleInnerMappingGroup}|hyperpage}{30}
\indexentry{InnerMappingGroup@\texttt {InnerMappingGroup}|hyperpage}{30}
\indexentry{LeftNucleus@\texttt {LeftNucleus}|hyperpage}{30}
\indexentry{MiddleNucleus@\texttt {MiddleNucleus}|hyperpage}{30}
\indexentry{RightNucleus@\texttt {RightNucleus}|hyperpage}{30}
\indexentry{Nuc@\texttt {Nuc}|hyperpage}{31}
\indexentry{NucleusOfQuasigroup@\texttt {NucleusOfQuasigroup}|hyperpage}{31}
\indexentry{NucleusOfLoop@\texttt {NucleusOfLoop}|hyperpage}{31}
\indexentry{Commutant@\texttt {Commutant}|hyperpage}{31}
\indexentry{Center@\texttt {Center}|hyperpage}{31}
\indexentry{AssociatorSubloop@\texttt {AssociatorSubloop}|hyperpage}{31}
\indexentry{IsNormal@\texttt {IsNormal}|hyperpage}{31}
\indexentry{subloop!normal|hyperpage}{31}
\indexentry{normal subloop|hyperpage}{31}
\indexentry{NormalClosure@\texttt {NormalClosure}|hyperpage}{31}
\indexentry{normal closure|hyperpage}{31}
\indexentry{IsSimple@\texttt {IsSimple}|hyperpage}{32}
\indexentry{simple loop|hyperpage}{32}
\indexentry{loop!simple|hyperpage}{32}
\indexentry{FactorLoop@\texttt {FactorLoop}|hyperpage}{32}
\indexentry{NaturalHomomorphismByNormalSubloop@\texttt {NaturalHomomorphismByNormalSubloop}|hyperpage}{32}
\indexentry{IsNilpotent@\texttt {IsNilpotent}|hyperpage}{32}
\indexentry{NilpotencyClassOfLoop@\texttt {NilpotencyClassOfLoop}|hyperpage}{32}
\indexentry{IsStronglyNilpotent@\texttt {IsStronglyNilpotent}|hyperpage}{32}
\indexentry{strongly nilpotent loop|hyperpage}{32}
\indexentry{nilpotent loop!strongly|hyperpage}{32}
\indexentry{loop!strongly nilpotent|hyperpage}{32}
\indexentry{UpperCentralSeries@\texttt {UpperCentralSeries}|hyperpage}{33}
\indexentry{LowerCentralSeries@\texttt {LowerCentralSeries}|hyperpage}{33}
\indexentry{central series!lower|hyperpage}{33}
\indexentry{IsSolvable@\texttt {IsSolvable}|hyperpage}{33}
\indexentry{DerivedSubloop@\texttt {DerivedSubloop}|hyperpage}{33}
\indexentry{DerivedLength@\texttt {DerivedLength}|hyperpage}{33}
\indexentry{FrattiniSubloop@\texttt {FrattiniSubloop}|hyperpage}{33}
\indexentry{Frattini subloop|hyperpage}{33}
\indexentry{FrattinifactorSize@\texttt {FrattinifactorSize}|hyperpage}{33}
\indexentry{IsomorphismQuasigroups@\texttt {IsomorphismQuasigroups}|hyperpage}{33}
\indexentry{IsomorphismLoops@\texttt {IsomorphismLoops}|hyperpage}{34}
\indexentry{QuasigroupsUpToIsomorphism@\texttt {QuasigroupsUpToIsomorphism}|hyperpage}{34}
\indexentry{LoopsUpToIsomorphism@\texttt {LoopsUpToIsomorphism}|hyperpage}{34}
\indexentry{AutomorphismGroup@\texttt {AutomorphismGroup}|hyperpage}{34}
\indexentry{IsomorphicCopyByPerm@\texttt {IsomorphicCopyByPerm}|hyperpage}{34}
\indexentry{IsomorphicCopyByNormalSubloop@\texttt {IsomorphicCopyByNormalSubloop}|hyperpage}{34}
\indexentry{Discriminator@\texttt {Discriminator}|hyperpage}{35}
\indexentry{AreEqualDiscriminators@\texttt {AreEqualDiscriminators}|hyperpage}{35}
\indexentry{IsotopismLoops@\texttt {IsotopismLoops}|hyperpage}{35}
\indexentry{LoopsUpToIsotopism@\texttt {LoopsUpToIsotopism}|hyperpage}{35}
\indexentry{IsAssociative@\texttt {IsAssociative}|hyperpage}{36}
\indexentry{IsCommutative@\texttt {IsCommutative}|hyperpage}{36}
\indexentry{IsPowerAssociative@\texttt {IsPowerAssociative}|hyperpage}{36}
\indexentry{quasigroup!power associative|hyperpage}{36}
\indexentry{power associative quasigroup|hyperpage}{36}
\indexentry{IsDiassociative@\texttt {IsDiassociative}|hyperpage}{36}
\indexentry{quasigroup!diassociative|hyperpage}{37}
\indexentry{diassociative quasigroup|hyperpage}{37}
\indexentry{inverse!left|hyperpage}{37}
\indexentry{inverse!right|hyperpage}{37}
\indexentry{HasLeftInverseProperty@\texttt {HasLeftInverseProperty}|hyperpage}{37}
\indexentry{HasRightInverseProperty@\texttt {HasRightInverseProperty}|hyperpage}{37}
\indexentry{HasInverseProperty@\texttt {HasInverseProperty}|hyperpage}{37}
\indexentry{inverse property!left|hyperpage}{37}
\indexentry{inverse property!right|hyperpage}{37}
\indexentry{inverse property|hyperpage}{37}
\indexentry{HasTwosidedInverses@\texttt {HasTwosidedInverses}|hyperpage}{37}
\indexentry{inverse!two-sided|hyperpage}{37}
\indexentry{HasWeakInverseProperty@\texttt {HasWeakInverseProperty}|hyperpage}{37}
\indexentry{inverse property!weak|hyperpage}{37}
\indexentry{HasAutomorphicInverseProperty@\texttt {HasAutomorphicInverseProperty}|hyperpage}{37}
\indexentry{automorphic inverse property|hyperpage}{37}
\indexentry{inverse property!automorphic|hyperpage}{37}
\indexentry{HasAntiautomorphicInverseProperty@\texttt {HasAntiautomorphicInverseProperty}|hyperpage}{37}
\indexentry{antiautomorphic inverse property|hyperpage}{37}
\indexentry{inverse property!antiautomorphic|hyperpage}{37}
\indexentry{IsSemisymmetric@\texttt {IsSemisymmetric}|hyperpage}{38}
\indexentry{semisymmetric quasigroup|hyperpage}{38}
\indexentry{quasigroup!semisymmetric|hyperpage}{38}
\indexentry{IsTotallySymmetric@\texttt {IsTotallySymmetric}|hyperpage}{38}
\indexentry{totally symmetric quasigroup|hyperpage}{38}
\indexentry{quasigroup!totally symmetric|hyperpage}{38}
\indexentry{IsIdempotent@\texttt {IsIdempotent}|hyperpage}{38}
\indexentry{idempotent quasigroup|hyperpage}{38}
\indexentry{quasigroup!idempotent|hyperpage}{38}
\indexentry{IsSteinerQuasigroup@\texttt {IsSteinerQuasigroup}|hyperpage}{38}
\indexentry{Steiner quasigroup|hyperpage}{38}
\indexentry{quasigroup!Steiner|hyperpage}{38}
\indexentry{unipotent quasigroup|hyperpage}{38}
\indexentry{quasigroup!unipotent|hyperpage}{38}
\indexentry{IsUnipotent@\texttt {IsUnipotent}|hyperpage}{38}
\indexentry{IsLeftDistributive@\texttt {IsLeftDistributive}|hyperpage}{38}
\indexentry{IsRightDistributive@\texttt {IsRightDistributive}|hyperpage}{38}
\indexentry{IsDistributive@\texttt {IsDistributive}|hyperpage}{38}
\indexentry{quasigroup!left distributive|hyperpage}{38}
\indexentry{distributive quasigroup!left|hyperpage}{38}
\indexentry{quasigroup!right distributive|hyperpage}{38}
\indexentry{distributive quasigroup!right|hyperpage}{38}
\indexentry{quasigroup!distributive|hyperpage}{38}
\indexentry{distributive quasigroup|hyperpage}{38}
\indexentry{IsEntropic@\texttt {IsEntropic}|hyperpage}{39}
\indexentry{IsMedial@\texttt {IsMedial}|hyperpage}{39}
\indexentry{entropic quasigroup|hyperpage}{39}
\indexentry{quasigroup!entropic|hyperpage}{39}
\indexentry{medial quasigroup|hyperpage}{39}
\indexentry{quasigroup!medial|hyperpage}{39}
\indexentry{loop!of Bol-Moufang type|hyperpage}{39}
\indexentry{identity!of Bol-Moufang type|hyperpage}{39}
\indexentry{alternative loop!left|hyperpage}{39}
\indexentry{loop!left alternative|hyperpage}{39}
\indexentry{alternative loop!right|hyperpage}{39}
\indexentry{loop!right alternative|hyperpage}{39}
\indexentry{nuclear square loop!left|hyperpage}{39}
\indexentry{loop!left nuclear square|hyperpage}{39}
\indexentry{nuclear square loop!middle|hyperpage}{39}
\indexentry{loop!middle nuclear square|hyperpage}{39}
\indexentry{nuclear square loop!right|hyperpage}{39}
\indexentry{loop!right nuclear square|hyperpage}{39}
\indexentry{flexible loop|hyperpage}{39}
\indexentry{loop!flexible|hyperpage}{39}
\indexentry{Bol loop!left|hyperpage}{39}
\indexentry{loop!left Bol|hyperpage}{39}
\indexentry{Bol loop!right|hyperpage}{39}
\indexentry{loop!right Bol|hyperpage}{39}
\indexentry{LC loop|hyperpage}{39}
\indexentry{loop!LC|hyperpage}{39}
\indexentry{RC loop|hyperpage}{39}
\indexentry{loop!RC|hyperpage}{39}
\indexentry{Moufang loop|hyperpage}{39}
\indexentry{loop!Moufang|hyperpage}{39}
\indexentry{C loop|hyperpage}{39}
\indexentry{loop!C|hyperpage}{39}
\indexentry{extra loop|hyperpage}{39}
\indexentry{loop!extra|hyperpage}{39}
\indexentry{alternative loop|hyperpage}{39}
\indexentry{loop!alternative|hyperpage}{39}
\indexentry{nuclear square loop|hyperpage}{39}
\indexentry{loop!nuclear square|hyperpage}{39}
\indexentry{IsExtraLoop@\texttt {IsExtraLoop}|hyperpage}{40}
\indexentry{IsMoufangLoop@\texttt {IsMoufangLoop}|hyperpage}{40}
\indexentry{IsCLoop@\texttt {IsCLoop}|hyperpage}{40}
\indexentry{IsLeftBolLoop@\texttt {IsLeftBolLoop}|hyperpage}{40}
\indexentry{IsRightBolLoop@\texttt {IsRightBolLoop}|hyperpage}{40}
\indexentry{IsLCLoop@\texttt {IsLCLoop}|hyperpage}{40}
\indexentry{IsRCLoop@\texttt {IsRCLoop}|hyperpage}{40}
\indexentry{IsLeftNuclearSquareLoop@\texttt {IsLeftNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsMiddleNuclearSquareLoop@\texttt {IsMiddleNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsRightNuclearSquareLoop@\texttt {IsRightNuclearSquareLoop}|hyperpage}{40}
\indexentry{IsNuclearSquareLoop@\texttt {IsNuclearSquareLoop}|hyperpage}{41}
\indexentry{IsFlexible@\texttt {IsFlexible}|hyperpage}{41}
\indexentry{IsLeftAlternative@\texttt {IsLeftAlternative}|hyperpage}{41}
\indexentry{IsRightAlternative@\texttt {IsRightAlternative}|hyperpage}{41}
\indexentry{IsAlternative@\texttt {IsAlternative}|hyperpage}{41}
\indexentry{power alternative loop!left|hyperpage}{42}
\indexentry{loop!left power alternative|hyperpage}{42}
\indexentry{power alternative loop!right|hyperpage}{42}
\indexentry{loop!right power alternative|hyperpage}{42}
\indexentry{power alternative loop|hyperpage}{42}
\indexentry{loop!power alternative|hyperpage}{42}
\indexentry{IsLeftPowerAlternative@\texttt {IsLeftPowerAlternative}|hyperpage}{42}
\indexentry{IsRightPowerAlternative@\texttt {IsRightPowerAlternative}|hyperpage}{42}
\indexentry{IsPowerAlternative@\texttt {IsPowerAlternative}|hyperpage}{42}
\indexentry{conjugacy closed loop!left|hyperpage}{42}
\indexentry{loop!left conjugacy closed|hyperpage}{42}
\indexentry{conjugacy closed loop!right|hyperpage}{42}
\indexentry{loop!right conjugacy closed|hyperpage}{42}
\indexentry{conjugacy closed loop|hyperpage}{42}
\indexentry{loop!conjugacy closed|hyperpage}{42}
\indexentry{IsLCCLoop@\texttt {IsLCCLoop}|hyperpage}{42}
\indexentry{IsLeftConjugacyClosedLoop@\texttt {IsLeftConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsRCCLoop@\texttt {IsRCCLoop}|hyperpage}{42}
\indexentry{IsRightConjugacyClosedLoop@\texttt {IsRightConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsCCLoop@\texttt {IsCCLoop}|hyperpage}{42}
\indexentry{IsConjugacyClosedLoop@\texttt {IsConjugacyClosedLoop}|hyperpage}{42}
\indexentry{IsOsbornLoop@\texttt {IsOsbornLoop}|hyperpage}{42}
\indexentry{Osborn loop|hyperpage}{43}
\indexentry{loop!Osborn|hyperpage}{43}
\indexentry{automorphic loop!left|hyperpage}{43}
\indexentry{loop!left automorphic|hyperpage}{43}
\indexentry{automorphic loop!middle|hyperpage}{43}
\indexentry{loop!middle automorphic|hyperpage}{43}
\indexentry{automorphic loop!right|hyperpage}{43}
\indexentry{loop!right automorphic|hyperpage}{43}
\indexentry{automorphic loop|hyperpage}{43}
\indexentry{loop!automorphic|hyperpage}{43}
\indexentry{IsLeftAutomorphicLoop@\texttt {IsLeftAutomorphicLoop}|hyperpage}{43}
\indexentry{IsLeftALoop@\texttt {IsLeftALoop}|hyperpage}{43}
\indexentry{IsMiddleAutomorphicLoop@\texttt {IsMiddleAutomorphicLoop}|hyperpage}{43}
\indexentry{IsMiddleALoop@\texttt {IsMiddleALoop}|hyperpage}{43}
\indexentry{IsRightAutomorphicLoop@\texttt {IsRightAutomorphicLoop}|hyperpage}{44}
\indexentry{IsRightALoop@\texttt {IsRightALoop}|hyperpage}{44}
\indexentry{IsAutomorphicLoop@\texttt {IsAutomorphicLoop}|hyperpage}{44}
\indexentry{IsALoop@\texttt {IsALoop}|hyperpage}{44}
\indexentry{IsCodeLoop@\texttt {IsCodeLoop}|hyperpage}{44}
\indexentry{code loop|hyperpage}{44}
\indexentry{loop!code|hyperpage}{44}
\indexentry{IsSteinerLoop@\texttt {IsSteinerLoop}|hyperpage}{44}
\indexentry{Steiner loop|hyperpage}{44}
\indexentry{loop!Steiner|hyperpage}{44}
\indexentry{IsLeftBruckLoop@\texttt {IsLeftBruckLoop}|hyperpage}{44}
\indexentry{IsLeftKLoop@\texttt {IsLeftKLoop}|hyperpage}{44}
\indexentry{Bruck loop!left|hyperpage}{44}
\indexentry{loop!left Bruck|hyperpage}{44}
\indexentry{K loop!left|hyperpage}{44}
\indexentry{loop!left K|hyperpage}{44}
\indexentry{IsRightBruckLoop@\texttt {IsRightBruckLoop}|hyperpage}{44}
\indexentry{IsRightKLoop@\texttt {IsRightKLoop}|hyperpage}{44}
\indexentry{Bruck loop!right|hyperpage}{44}
\indexentry{loop!right Bruck|hyperpage}{44}
\indexentry{K loop!right|hyperpage}{44}
\indexentry{loop!right K|hyperpage}{44}
\indexentry{AssociatedLeftBruckLoop@\texttt {AssociatedLeftBruckLoop}|hyperpage}{45}
\indexentry{AssociatedRightBruckLoop@\texttt {AssociatedRightBruckLoop}|hyperpage}{45}
\indexentry{loop!left Bol|hyperpage}{45}
\indexentry{Bol loop!left|hyperpage}{45}
\indexentry{Bruck loop!associated left|hyperpage}{45}
\indexentry{loop!associated left Bruck|hyperpage}{45}
\indexentry{IsExactGroupFactorization@\texttt {IsExactGroupFactorization}|hyperpage}{45}
\indexentry{exact group factorization|hyperpage}{45}
\indexentry{RightBolLoopByExactGroupFactorization@\texttt {Right}\discretionary {-}{}{}\texttt {Bol}\discretionary {-}{}{}\texttt {Loop}\discretionary {-}{}{}\texttt {By}\discretionary {-}{}{}\texttt {Exact}\discretionary {-}{}{}\texttt {Group}\discretionary {-}{}{}\texttt {Factorization}|hyperpage}{45}
\indexentry{modification!Moufang|hyperpage}{46}
\indexentry{LoopByCyclicModification@\texttt {LoopByCyclicModification}|hyperpage}{46}
\indexentry{modification!cyclic|hyperpage}{46}
\indexentry{LoopByDihedralModification@\texttt {LoopByDihedralModification}|hyperpage}{46}
\indexentry{modification!dihedral|hyperpage}{46}
\indexentry{LoopMG2@\texttt {LoopMG2}|hyperpage}{46}
\indexentry{Chein loop|hyperpage}{46}
\indexentry{loop!Chein|hyperpage}{46}
\indexentry{group with triality|hyperpage}{46}
\indexentry{TrialityPermGroup@\texttt {TrialityPermGroup}|hyperpage}{47}
\indexentry{TrialityPcGroup@\texttt {TrialityPcGroup}|hyperpage}{47}
\indexentry{AllLoopTablesInGroup@\texttt {AllLoopTablesInGroup}|hyperpage}{47}
\indexentry{AllProperLoopTablesInGroup@\texttt {AllProperLoopTablesInGroup}|hyperpage}{47}
\indexentry{OneLoopTableInGroup@\texttt {OneLoopTableInGroup}|hyperpage}{47}
\indexentry{OneProperLoopTableInGroup@\texttt {OneProperLoopTableInGroup}|hyperpage}{48}
\indexentry{AllLoopsWithMltGroup@\texttt {AllLoopsWithMltGroup}|hyperpage}{48}
\indexentry{OneLoopWithMltGroup@\texttt {OneLoopWithMltGroup}|hyperpage}{48}
\indexentry{LibraryLoop@\texttt {LibraryLoop}|hyperpage}{49}
\indexentry{MyLibraryLoop@\texttt {MyLibraryLoop}|hyperpage}{49}
\indexentry{DisplayLibraryInfo@\texttt {DisplayLibraryInfo}|hyperpage}{50}
\indexentry{LeftBolLoop@\texttt {LeftBolLoop}|hyperpage}{50}
\indexentry{RightBolLoop@\texttt {RightBolLoop}|hyperpage}{50}
\indexentry{MoufangLoop@\texttt {MoufangLoop}|hyperpage}{50}
\indexentry{octonion loop|hyperpage}{50}
\indexentry{loop!octonion|hyperpage}{50}
\indexentry{CodeLoop@\texttt {CodeLoop}|hyperpage}{51}
\indexentry{SteinerLoop@\texttt {SteinerLoop}|hyperpage}{51}
\indexentry{RCCLoop@\texttt {RCCLoop}|hyperpage}{52}
\indexentry{RightConjugacyClosedLoop@\texttt {RightConjugacyClosedLoop}|hyperpage}{52}
\indexentry{LCCLoop@\texttt {LCCLoop}|hyperpage}{52}
\indexentry{LeftConjugacyClosedLoop@\texttt {LeftConjugacyClosedLoop}|hyperpage}{52}
\indexentry{CCLoop@\texttt {CCLoop}|hyperpage}{52}
\indexentry{ConjugacyClosedLoop@\texttt {ConjugacyClosedLoop}|hyperpage}{52}
\indexentry{SmallLoop@\texttt {SmallLoop}|hyperpage}{53}
\indexentry{Paige loop|hyperpage}{53}
\indexentry{loop!Paige|hyperpage}{53}
\indexentry{PaigeLoop@\texttt {PaigeLoop}|hyperpage}{53}
\indexentry{NilpotentLoop@\texttt {NilpotentLoop}|hyperpage}{53}
\indexentry{AutomorphicLoop@\texttt {AutomorphicLoop}|hyperpage}{53}
\indexentry{sedenion loop|hyperpage}{54}
\indexentry{loop!sedenion|hyperpage}{54}
\indexentry{InterestingLoop@\texttt {InterestingLoop}|hyperpage}{54}
\indexentry{ItpSmallLoop@\texttt {ItpSmallLoop}|hyperpage}{54}

6
doc/loops.ilg
View File

@ -1,6 +0,0 @@
This is makeindex, version 2.15 [MiKTeX 2.9 64-bit] (kpathsea + Thai support).
Scanning input file loops.idx....done (427 entries accepted, 0 rejected).
Sorting entries......done (4056 comparisons).
Generating output file loops.ind....done (485 lines written, 0 warnings).
Output written in loops.ind.
Transcript written in loops.ilg.

485
doc/loops.ind
View File

@ -1,485 +0,0 @@
\begin{theindex}
\item \texttt {AllLoopsWithMltGroup}, \hyperpage{48}
\item \texttt {AllLoopTablesInGroup}, \hyperpage{47}
\item \texttt {AllProperLoopTablesInGroup}, \hyperpage{47}
\item \texttt {AllSubloops}, \hyperpage{28}
\item \texttt {AllSubquasigroups}, \hyperpage{27}
\item alternative loop, \hyperpage{39}
\subitem left, \hyperpage{39}
\subitem right, \hyperpage{39}
\item antiautomorphic inverse property, \hyperpage{37}
\item \texttt {AreEqualDiscriminators}, \hyperpage{35}
\item \texttt {AssociatedLeftBruckLoop}, \hyperpage{45}
\item \texttt {AssociatedRightBruckLoop}, \hyperpage{45}
\item \texttt {Associator}, \hyperpage{24}
\item associator, \hyperpage{9}
\item associator subloop, \hyperpage{9}
\item \texttt {AssociatorSubloop}, \hyperpage{31}
\item automorphic inverse property, \hyperpage{37}
\item automorphic loop, \hyperpage{43}
\subitem left, \hyperpage{43}
\subitem middle, \hyperpage{43}
\subitem right, \hyperpage{43}
\item \texttt {AutomorphicLoop}, \hyperpage{53}
\item \texttt {AutomorphismGroup}, \hyperpage{34}
\indexspace
\item Bol loop
\subitem left, \hyperpage{12}, \hyperpage{39}, \hyperpage{45}
\subitem right, \hyperpage{39}
\item Bruck loop
\subitem associated left, \hyperpage{45}
\subitem left, \hyperpage{44}
\subitem right, \hyperpage{44}
\indexspace
\item C loop, \hyperpage{39}
\item \texttt {CanonicalCayleyTable}, \hyperpage{15}
\item \texttt {CanonicalCopy}, \hyperpage{15}
\item Cayley table, \hyperpage{14}
\subitem canonical, \hyperpage{15}
\item \texttt {CayleyTable}, \hyperpage{22}
\item \texttt {CayleyTableByPerms}, \hyperpage{17}
\item \texttt {CCLoop}, \hyperpage{52}
\item \texttt {Center}, \hyperpage{31}
\item center, \hyperpage{9}
\item central series
\subitem lower, \hyperpage{33}
\subitem upper, \hyperpage{9}
\item Chein loop, \hyperpage{46}
\item cocycle, \hyperpage{18}
\item code loop, \hyperpage{44}
\item \texttt {CodeLoop}, \hyperpage{51}
\item \texttt {Commutant}, \hyperpage{31}
\item commutant, \hyperpage{9}
\item \texttt {Commutator}, \hyperpage{24}
\item commutator, \hyperpage{9}
\item conjugacy closed loop, \hyperpage{42}
\subitem left, \hyperpage{42}
\subitem right, \hyperpage{42}
\item \texttt {ConjugacyClosedLoop}, \hyperpage{52}
\item conjugation, \hyperpage{30}
\item coset, \hyperpage{28}
\indexspace
\item derived series, \hyperpage{9}
\item derived subloop, \hyperpage{9}
\item \texttt {DerivedLength}, \hyperpage{33}
\item \texttt {DerivedSubloop}, \hyperpage{33}
\item diassociative quasigroup, \hyperpage{37}
\item \texttt {DirectProduct}, \hyperpage{21}
\item \texttt {Discriminator}, \hyperpage{35}
\item \texttt {DisplayLibraryInfo}, \hyperpage{50}
\item distributive quasigroup, \hyperpage{38}
\subitem left, \hyperpage{38}
\subitem right, \hyperpage{38}
\item division
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\indexspace
\item \texttt {Elements}, \hyperpage{22}
\item entropic quasigroup, \hyperpage{39}
\item exact group factorization, \hyperpage{45}
\item \texttt {Exponent}, \hyperpage{23}
\item exponent, \hyperpage{23}
\item extension, \hyperpage{18}
\subitem nuclear, \hyperpage{18}
\item extra loop, \hyperpage{39}
\indexspace
\item \texttt {FactorLoop}, \hyperpage{32}
\item flexible loop, \hyperpage{39}
\item folder
\subitem quasigroup, \hyperpage{18}
\item Frattini subloop, \hyperpage{33}
\item \texttt {FrattinifactorSize}, \hyperpage{33}
\item \texttt {FrattiniSubloop}, \hyperpage{33}
\indexspace
\item \texttt {GeneratorsOfLoop}, \hyperpage{24}
\item \texttt {GeneratorsOfQuasigroup}, \hyperpage{24}
\item \texttt {GeneratorsSmallest}, \hyperpage{25}
\item group, \hyperpage{8}
\item group with triality, \hyperpage{46}
\item groupoid, \hyperpage{8}
\indexspace
\item \texttt {HasAntiautomorphicInverseProperty}, \hyperpage{37}
\item \texttt {HasAutomorphicInverseProperty}, \hyperpage{37}
\item \texttt {HasInverseProperty}, \hyperpage{37}
\item \texttt {HasLeftInverseProperty}, \hyperpage{37}
\item \texttt {HasRightInverseProperty}, \hyperpage{37}
\item \texttt {HasTwosidedInverses}, \hyperpage{37}
\item \texttt {HasWeakInverseProperty}, \hyperpage{37}
\item homomorphism, \hyperpage{9}
\item homotopism, \hyperpage{10}
\indexspace
\item idempotent quasigroup, \hyperpage{38}
\item identity
\subitem element, \hyperpage{8}
\subitem of Bol-Moufang type, \hyperpage{39}
\item inner mapping
\subitem left, \hyperpage{29}
\subitem middle, \hyperpage{30}
\subitem right, \hyperpage{29}
\item inner mapping group, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem middle, \hyperpage{30}
\subitem right, \hyperpage{9}
\item \texttt {InnerMappingGroup}, \hyperpage{30}
\item \texttt {InterestingLoop}, \hyperpage{54}
\item \texttt {IntoGroup}, \hyperpage{20}
\item \texttt {IntoLoop}, \hyperpage{20}
\item \texttt {IntoQuasigroup}, \hyperpage{20}
\item \texttt {Inverse}, \hyperpage{24}
\item inverse, \hyperpage{24}
\subitem left, \hyperpage{24}, \hyperpage{37}
\subitem right, \hyperpage{24}, \hyperpage{37}
\subitem two-sided, \hyperpage{8}, \hyperpage{37}
\item inverse property, \hyperpage{37}
\subitem antiautomorphic, \hyperpage{37}
\subitem automorphic, \hyperpage{37}
\subitem left, \hyperpage{37}
\subitem right, \hyperpage{37}
\subitem weak, \hyperpage{37}
\item \texttt {IsALoop}, \hyperpage{44}
\item \texttt {IsAlternative}, \hyperpage{41}
\item \texttt {IsAssociative}, \hyperpage{36}
\item \texttt {IsAutomorphicLoop}, \hyperpage{44}
\item \texttt {IsCCLoop}, \hyperpage{42}
\item \texttt {IsCLoop}, \hyperpage{40}
\item \texttt {IsCodeLoop}, \hyperpage{44}
\item \texttt {IsCommutative}, \hyperpage{36}
\item \texttt {IsConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsDiassociative}, \hyperpage{36}
\item \texttt {IsDistributive}, \hyperpage{38}
\item \texttt {IsEntropic}, \hyperpage{39}
\item \texttt {IsExactGroupFactorization}, \hyperpage{45}
\item \texttt {IsExtraLoop}, \hyperpage{40}
\item \texttt {IsFlexible}, \hyperpage{41}
\item \texttt {IsIdempotent}, \hyperpage{38}
\item \texttt {IsLCCLoop}, \hyperpage{42}
\item \texttt {IsLCLoop}, \hyperpage{40}
\item \texttt {IsLeftALoop}, \hyperpage{43}
\item \texttt {IsLeftAlternative}, \hyperpage{41}
\item \texttt {IsLeftAutomorphicLoop}, \hyperpage{43}
\item \texttt {IsLeftBolLoop}, \hyperpage{40}
\item \texttt {IsLeftBruckLoop}, \hyperpage{44}
\item \texttt {IsLeftConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsLeftDistributive}, \hyperpage{38}
\item \texttt {IsLeftKLoop}, \hyperpage{44}
\item \texttt {IsLeftNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsLeftPowerAlternative}, \hyperpage{42}
\item IsLoop, \hyperpage{11}
\item \texttt {IsLoopCayleyTable}, \hyperpage{14}
\item IsLoopElement, \hyperpage{11}
\item \texttt {IsLoopTable}, \hyperpage{14}
\item \texttt {IsMedial}, \hyperpage{39}
\item \texttt {IsMiddleALoop}, \hyperpage{43}
\item \texttt {IsMiddleAutomorphicLoop}, \hyperpage{43}
\item \texttt {IsMiddleNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsMoufangLoop}, \hyperpage{40}
\item \texttt {IsNilpotent}, \hyperpage{32}
\item \texttt {IsNormal}, \hyperpage{31}
\item \texttt {IsNuclearSquareLoop}, \hyperpage{41}
\item \texttt {IsomorphicCopyByNormalSubloop}, \hyperpage{34}
\item \texttt {IsomorphicCopyByPerm}, \hyperpage{34}
\item isomorphism, \hyperpage{9}
\item \texttt {IsomorphismLoops}, \hyperpage{34}
\item \texttt {IsomorphismQuasigroups}, \hyperpage{33}
\item \texttt {IsOsbornLoop}, \hyperpage{42}
\item isotopism, \hyperpage{10}
\subitem principal, \hyperpage{10}
\item \texttt {IsotopismLoops}, \hyperpage{35}
\item \texttt {IsPowerAlternative}, \hyperpage{42}
\item \texttt {IsPowerAssociative}, \hyperpage{36}
\item IsQuasigroup, \hyperpage{11}
\item \texttt {IsQuasigroupCayleyTable}, \hyperpage{14}
\item IsQuasigroupElement, \hyperpage{11}
\item \texttt {IsQuasigroupTable}, \hyperpage{14}
\item \texttt {IsRCCLoop}, \hyperpage{42}
\item \texttt {IsRCLoop}, \hyperpage{40}
\item \texttt {IsRightALoop}, \hyperpage{44}
\item \texttt {IsRightAlternative}, \hyperpage{41}
\item \texttt {IsRightAutomorphicLoop}, \hyperpage{44}
\item \texttt {IsRightBolLoop}, \hyperpage{40}
\item \texttt {IsRightBruckLoop}, \hyperpage{44}
\item \texttt {IsRightConjugacyClosedLoop}, \hyperpage{42}
\item \texttt {IsRightDistributive}, \hyperpage{38}
\item \texttt {IsRightKLoop}, \hyperpage{44}
\item \texttt {IsRightNuclearSquareLoop}, \hyperpage{40}
\item \texttt {IsRightPowerAlternative}, \hyperpage{42}
\item \texttt {IsSemisymmetric}, \hyperpage{38}
\item \texttt {IsSimple}, \hyperpage{32}
\item \texttt {IsSolvable}, \hyperpage{33}
\item \texttt {IsSteinerLoop}, \hyperpage{44}
\item \texttt {IsSteinerQuasigroup}, \hyperpage{38}
\item \texttt {IsStronglyNilpotent}, \hyperpage{32}
\item \texttt {IsSubloop}, \hyperpage{27}
\item \texttt {IsSubquasigroup}, \hyperpage{27}
\item \texttt {IsTotallySymmetric}, \hyperpage{38}
\item \texttt {IsUnipotent}, \hyperpage{38}
\item \texttt {ItpSmallLoop}, \hyperpage{54}
\indexspace
\item K loop
\subitem left, \hyperpage{44}
\subitem right, \hyperpage{44}
\indexspace
\item latin square, \hyperpage{8}, \hyperpage{14}
\subitem random, \hyperpage{19}
\item LC loop, \hyperpage{39}
\item \texttt {LCCLoop}, \hyperpage{52}
\item \texttt {LeftBolLoop}, \hyperpage{50}
\item \texttt {LeftConjugacyClosedLoop}, \hyperpage{52}
\item \texttt {LeftDivision}, \hyperpage{23}
\item \texttt {LeftDivisionCayleyTable}, \hyperpage{23}
\item \texttt {LeftInnerMapping}, \hyperpage{30}
\item \texttt {LeftInnerMappingGroup}, \hyperpage{30}
\item \texttt {LeftInverse}, \hyperpage{24}
\item \texttt {LeftMultiplicationGroup}, \hyperpage{29}
\item \texttt {LeftNucleus}, \hyperpage{30}
\item \texttt {LeftSection}, \hyperpage{28}
\item \texttt {LeftTranslation}, \hyperpage{28}
\item \texttt {LibraryLoop}, \hyperpage{49}
\item loop, \hyperpage{8}
\subitem alternative, \hyperpage{39}
\subitem associated left Bruck, \hyperpage{45}
\subitem automorphic, \hyperpage{43}
\subitem C, \hyperpage{39}
\subitem Chein, \hyperpage{46}
\subitem code, \hyperpage{44}
\subitem conjugacy closed, \hyperpage{42}
\subitem extra, \hyperpage{39}
\subitem flexible, \hyperpage{39}
\subitem LC, \hyperpage{39}
\subitem left alternative, \hyperpage{39}
\subitem left automorphic, \hyperpage{43}
\subitem left Bol, \hyperpage{12}, \hyperpage{39}, \hyperpage{45}
\subitem left Bruck, \hyperpage{44}
\subitem left conjugacy closed, \hyperpage{42}
\subitem left K, \hyperpage{44}
\subitem left nuclear square, \hyperpage{39}
\subitem left power alternative, \hyperpage{42}
\subitem middle automorphic, \hyperpage{43}
\subitem middle nuclear square, \hyperpage{39}
\subitem Moufang, \hyperpage{39}
\subitem nilpotent, \hyperpage{9}, \hyperpage{19}
\subitem nuclear square, \hyperpage{39}
\subitem octonion, \hyperpage{50}
\subitem of Bol-Moufang type, \hyperpage{39}
\subitem Osborn, \hyperpage{43}
\subitem Paige, \hyperpage{53}
\subitem power alternative, \hyperpage{42}
\subitem power associative, \hyperpage{23}
\subitem RC, \hyperpage{39}
\subitem right alternative, \hyperpage{39}
\subitem right automorphic, \hyperpage{43}
\subitem right Bol, \hyperpage{39}
\subitem right Bruck, \hyperpage{44}
\subitem right conjugacy closed, \hyperpage{42}
\subitem right K, \hyperpage{44}
\subitem right nuclear square, \hyperpage{39}
\subitem right power alternative, \hyperpage{42}
\subitem sedenion, \hyperpage{54}
\subitem simple, \hyperpage{12}, \hyperpage{32}
\subitem solvable, \hyperpage{9}
\subitem Steiner, \hyperpage{44}
\subitem strongly nilpotent, \hyperpage{32}
\item loop isotope
\subitem principal, \hyperpage{10}
\item loop table, \hyperpage{14}
\item \texttt {LoopByCayleyTable}, \hyperpage{15}
\item \texttt {LoopByCyclicModification}, \hyperpage{46}
\item \texttt {LoopByDihedralModification}, \hyperpage{46}
\item \texttt {LoopByExtension}, \hyperpage{18}
\item \texttt {LoopByLeftSection}, \hyperpage{17}
\item \texttt {LoopByRightFolder}, \hyperpage{18}
\item \texttt {LoopByRightSection}, \hyperpage{17}
\item \texttt {LoopFromFile}, \hyperpage{17}
\item \texttt {LoopMG2}, \hyperpage{46}
\item \texttt {LoopsUpToIsomorphism}, \hyperpage{34}
\item \texttt {LoopsUpToIsotopism}, \hyperpage{35}
\item \texttt {LowerCentralSeries}, \hyperpage{33}
\indexspace
\item magma, \hyperpage{8}
\item medial quasigroup, \hyperpage{39}
\item \texttt {MiddleInnerMapping}, \hyperpage{30}
\item \texttt {MiddleInnerMappingGroup}, \hyperpage{30}
\item \texttt {MiddleNucleus}, \hyperpage{30}
\item modification
\subitem cyclic, \hyperpage{46}
\subitem dihedral, \hyperpage{46}
\subitem Moufang, \hyperpage{46}
\item Moufang loop, \hyperpage{39}
\item \texttt {MoufangLoop}, \hyperpage{50}
\item multiplication group, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem relative, \hyperpage{29}
\subitem relative left, \hyperpage{29}
\subitem relative right , \hyperpage{29}
\subitem right, \hyperpage{9}
\item multiplication table, \hyperpage{14}
\item \texttt {MultiplicationGroup}, \hyperpage{29}
\item \texttt {MyLibraryLoop}, \hyperpage{49}
\indexspace
\item \texttt {NaturalHomomorphismByNormalSubloop}, \hyperpage{32}
\item neutral element, \hyperpage{8}
\item nilpotence class, \hyperpage{9}
\item \texttt {NilpotencyClassOfLoop}, \hyperpage{32}
\item nilpotent loop, \hyperpage{9}
\subitem strongly, \hyperpage{32}
\item \texttt {NilpotentLoop}, \hyperpage{53}
\item normal closure, \hyperpage{31}
\item normal subloop, \hyperpage{31}
\item \texttt {NormalClosure}, \hyperpage{31}
\item \texttt {NormalizedQuasigroupTable}, \hyperpage{15}
\item \texttt {Nuc}, \hyperpage{31}
\item nuclear square loop, \hyperpage{39}
\subitem left, \hyperpage{39}
\subitem middle, \hyperpage{39}
\subitem right, \hyperpage{39}
\item \texttt {NuclearExtension}, \hyperpage{18}
\item nucleus, \hyperpage{9}
\subitem left, \hyperpage{9}
\subitem middle, \hyperpage{9}
\subitem right, \hyperpage{9}
\item \texttt {NucleusOfLoop}, \hyperpage{31}
\item \texttt {NucleusOfQuasigroup}, \hyperpage{31}
\indexspace
\item octonion loop, \hyperpage{50}
\item \texttt {One}, \hyperpage{22}
\item \texttt {OneLoopTableInGroup}, \hyperpage{47}
\item \texttt {OneLoopWithMltGroup}, \hyperpage{48}
\item \texttt {OneProperLoopTableInGroup}, \hyperpage{48}
\item \texttt {Opposite}, \hyperpage{21}
\item opposite quasigroup, \hyperpage{21}
\item \texttt {OppositeLoop}, \hyperpage{21}
\item \texttt {OppositeQuasigroup}, \hyperpage{21}
\item Osborn loop, \hyperpage{43}
\indexspace
\item Paige loop, \hyperpage{53}
\item \texttt {PaigeLoop}, \hyperpage{53}
\item \texttt {Parent}, \hyperpage{26}
\item \texttt {PosInParent}, \hyperpage{27}
\item \texttt {Position}, \hyperpage{26}
\item power alternative loop, \hyperpage{42}
\subitem left, \hyperpage{42}
\subitem right, \hyperpage{42}
\item power associative loop, \hyperpage{23}
\item power associative quasigroup, \hyperpage{36}
\item \texttt {PrincipalLoopIsotope}, \hyperpage{20}
\indexspace
\item quasigroup, \hyperpage{8}
\subitem diassociative, \hyperpage{37}
\subitem distributive, \hyperpage{38}
\subitem entropic, \hyperpage{39}
\subitem idempotent, \hyperpage{38}
\subitem left distributive, \hyperpage{38}
\subitem medial, \hyperpage{39}
\subitem opposite, \hyperpage{21}
\subitem power associative, \hyperpage{36}
\subitem right distributive, \hyperpage{38}
\subitem semisymmetric, \hyperpage{38}
\subitem Steiner, \hyperpage{38}
\subitem totally symmetric, \hyperpage{38}
\subitem unipotent, \hyperpage{38}
\item quasigroup table, \hyperpage{14}
\item \texttt {QuasigroupByCayleyTable}, \hyperpage{15}
\item \texttt {QuasigroupByLeftSection}, \hyperpage{17}
\item \texttt {QuasigroupByRightFolder}, \hyperpage{18}
\item \texttt {QuasigroupByRightSection}, \hyperpage{17}
\item \texttt {QuasigroupFromFile}, \hyperpage{17}
\item \texttt {QuasigroupsUpToIsomorphism}, \hyperpage{34}
\indexspace
\item \texttt {RandomLoop}, \hyperpage{19}
\item \texttt {RandomNilpotentLoop}, \hyperpage{19}
\item \texttt {RandomQuasigroup}, \hyperpage{19}
\item RC loop, \hyperpage{39}
\item \texttt {RCCLoop}, \hyperpage{52}
\item \texttt {RelativeLeftMultiplicationGroup}, \hyperpage{29}
\item \texttt {RelativeMultiplicationGroup}, \hyperpage{29}
\item \texttt {RelativeRightMultiplicationGroup}, \hyperpage{29}
\item \texttt {RightBolLoop}, \hyperpage{50}
\item \texttt {Right}\discretionary {-}{}{}\texttt {Bol}\discretionary {-}{}{}\texttt {Loop}\discretionary {-}{}{}\texttt {By}\discretionary {-}{}{}\texttt {Exact}\discretionary {-}{}{}\texttt {Group}\discretionary {-}{}{}\texttt {Factorization},
\hyperpage{45}
\item \texttt {RightConjugacyClosedLoop}, \hyperpage{52}
\item \texttt {RightCosets}, \hyperpage{28}
\item \texttt {RightDivision}, \hyperpage{23}
\item \texttt {RightDivisionCayleyTable}, \hyperpage{23}
\item \texttt {RightInnerMapping}, \hyperpage{30}
\item \texttt {RightInnerMappingGroup}, \hyperpage{30}
\item \texttt {RightInverse}, \hyperpage{24}
\item \texttt {RightMultiplicationGroup}, \hyperpage{29}
\item \texttt {RightNucleus}, \hyperpage{30}
\item \texttt {RightSection}, \hyperpage{28}
\item \texttt {RightTranslation}, \hyperpage{28}
\item \texttt {RightTransversal}, \hyperpage{28}
\indexspace
\item section
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\item sedenion loop, \hyperpage{54}
\item semisymmetric quasigroup, \hyperpage{38}
\item \texttt {SetLoopElmName}, \hyperpage{13}
\item \texttt {SetQuasigroupElmName}, \hyperpage{13}
\item simple loop, \hyperpage{12}, \hyperpage{32}
\item \texttt {Size}, \hyperpage{22}
\item \texttt {SmallGeneratingSet}, \hyperpage{25}
\item \texttt {SmallLoop}, \hyperpage{53}
\item solvability class, \hyperpage{9}
\item solvable loop, \hyperpage{9}
\item Steiner loop, \hyperpage{44}
\item Steiner quasigroup, \hyperpage{38}
\item \texttt {SteinerLoop}, \hyperpage{51}
\item strongly nilpotent loop, \hyperpage{32}
\item \texttt {Subloop}, \hyperpage{27}
\item subloop, \hyperpage{9}
\subitem normal, \hyperpage{9}, \hyperpage{31}
\item \texttt {Subquasigroup}, \hyperpage{27}
\item subquasigroup, \hyperpage{9}
\indexspace
\item totally symmetric quasigroup, \hyperpage{38}
\item translation
\subitem left, \hyperpage{8}
\subitem right, \hyperpage{8}
\item transversal, \hyperpage{28}
\item \texttt {TrialityPcGroup}, \hyperpage{47}
\item \texttt {TrialityPermGroup}, \hyperpage{47}
\indexspace
\item unipotent quasigroup, \hyperpage{38}
\item \texttt {UpperCentralSeries}, \hyperpage{33}
\end{theindex}

856
doc/loops.log
View File

@ -1,856 +0,0 @@
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (MiKTeX 2.9 64-bit) (preloaded format=latex 2014.9.18) 27 OCT 2016 11:38
entering extended mode
**loops.tex
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.tex
LaTeX2e <2011/06/27>
Babel <v3.8m> and hyphenation patterns for english, afrikaans, ancientgreek, ar
abic, armenian, assamese, basque, bengali, bokmal, bulgarian, catalan, coptic,
croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga
lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland
ic, indonesian, interlingua, irish, italian, kannada, kurmanji, latin, latvian,
lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n
german-x-2013-05-26, nynorsk, oriya, panjabi, pinyin, polish, portuguese, roman
ian, russian, sanskrit, serbian, slovak, slovenian, spanish, swedish, swissgerm
an, tamil, telugu, turkish, turkmen, ukenglish, ukrainian, uppersorbian, usengl
ishmax, welsh, loaded.
("C:\Program Files\MiKTeX 2.9\tex\latex\base\report.cls"
Document Class: report 2007/10/19 v1.4h Standard LaTeX document class
("C:\Program Files\MiKTeX 2.9\tex\latex\base\size11.clo"
File: size11.clo 2007/10/19 v1.4h Standard LaTeX file (size option)
)
\c@part=\count79
\c@chapter=\count80
\c@section=\count81
\c@subsection=\count82
\c@subsubsection=\count83
\c@paragraph=\count84
\c@subparagraph=\count85
\c@figure=\count86
\c@table=\count87
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
)
("C:\Program Files\MiKTeX 2.9\tex\latex\a4wide\a4wide.sty"
Package: a4wide 1994/08/30
("C:\Program Files\MiKTeX 2.9\tex\latex\ntgclass\a4.sty"
Package: a4 2004/04/15 v1.2g A4 based page layout
))
("C:\Program Files\MiKTeX 2.9\tex\latex\amsfonts\amssymb.sty"
Package: amssymb 2013/01/14 v3.01 AMS font symbols
("C:\Program Files\MiKTeX 2.9\tex\latex\amsfonts\amsfonts.sty"
Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
\@emptytoks=\toks14
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
(Font) U/euf/m/n --> U/euf/b/n on input line 106.
))
("C:\Program Files\MiKTeX 2.9\tex\latex\base\inputenc.sty"
Package: inputenc 2008/03/30 v1.1d Input encoding file
\inpenc@prehook=\toks15
\inpenc@posthook=\toks16
("C:\Program Files\MiKTeX 2.9\tex\latex\base\latin1.def"
File: latin1.def 2008/03/30 v1.1d Input encoding file
))
("C:\Program Files\MiKTeX 2.9\tex\latex\base\makeidx.sty"
Package: makeidx 2000/03/29 v1.0m Standard LaTeX package
)
\@indexfile=\write3
Writing index file loops.idx
("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\color.sty"
Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC)
("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\color.cfg"
File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive
)
Package color Info: Driver file: dvips.def on input line 130.
("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\dvips.def"
File: dvips.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR)
)
("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\dvipsnam.def"
File: dvipsnam.def 1999/02/16 v3.0i Driver-dependant file (DPC,SPQR)
))
("C:\Program Files\MiKTeX 2.9\tex\latex\fancyvrb\fancyvrb.sty"
Package: fancyvrb 2008/02/07
Style option: `fancyvrb' v2.7a, with DG/SPQR fixes, and firstline=lastline fix
<2008/02/07> (tvz) ("C:\Program Files\MiKTeX 2.9\tex\latex\graphics\keyval.sty"
Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
\KV@toks@=\toks17
)
\FV@CodeLineNo=\count88
\FV@InFile=\read1
\FV@TabBox=\box26
\c@FancyVerbLine=\count89
\FV@StepNumber=\count90
\FV@OutFile=\write4
) ("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\mathptmx.sty"
Package: mathptmx 2005/04/12 PSNFSS-v9.2a Times w/ Math, improved (SPQR, WaS)
LaTeX Font Info: Redeclaring symbol font `operators' on input line 28.
LaTeX Font Info: Overwriting symbol font `operators' in version `normal'
(Font) OT1/cmr/m/n --> OT1/ztmcm/m/n on input line 28.
LaTeX Font Info: Overwriting symbol font `operators' in version `bold'
(Font) OT1/cmr/bx/n --> OT1/ztmcm/m/n on input line 28.
LaTeX Font Info: Redeclaring symbol font `letters' on input line 29.
LaTeX Font Info: Overwriting symbol font `letters' in version `normal'
(Font) OML/cmm/m/it --> OML/ztmcm/m/it on input line 29.
LaTeX Font Info: Overwriting symbol font `letters' in version `bold'
(Font) OML/cmm/b/it --> OML/ztmcm/m/it on input line 29.
LaTeX Font Info: Redeclaring symbol font `symbols' on input line 30.
LaTeX Font Info: Overwriting symbol font `symbols' in version `normal'
(Font) OMS/cmsy/m/n --> OMS/ztmcm/m/n on input line 30.
LaTeX Font Info: Overwriting symbol font `symbols' in version `bold'
(Font) OMS/cmsy/b/n --> OMS/ztmcm/m/n on input line 30.
LaTeX Font Info: Redeclaring symbol font `largesymbols' on input line 31.
LaTeX Font Info: Overwriting symbol font `largesymbols' in version `normal'
(Font) OMX/cmex/m/n --> OMX/ztmcm/m/n on input line 31.
LaTeX Font Info: Overwriting symbol font `largesymbols' in version `bold'
(Font) OMX/cmex/m/n --> OMX/ztmcm/m/n on input line 31.
\symbold=\mathgroup6
\symitalic=\mathgroup7
LaTeX Font Info: Redeclaring math alphabet \mathbf on input line 34.
LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `normal'
(Font) OT1/cmr/bx/n --> OT1/ptm/bx/n on input line 34.
LaTeX Font Info: Overwriting math alphabet `\mathbf' in version `bold'
(Font) OT1/cmr/bx/n --> OT1/ptm/bx/n on input line 34.
LaTeX Font Info: Redeclaring math alphabet \mathit on input line 35.
LaTeX Font Info: Overwriting math alphabet `\mathit' in version `normal'
(Font) OT1/cmr/m/it --> OT1/ptm/m/it on input line 35.
LaTeX Font Info: Overwriting math alphabet `\mathit' in version `bold'
(Font) OT1/cmr/bx/it --> OT1/ptm/m/it on input line 35.
LaTeX Info: Redefining \hbar on input line 50.
)
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\helvet.sty"
Package: helvet 2005/04/12 PSNFSS-v9.2a (WaS)
)
("C:\Program Files\MiKTeX 2.9\tex\latex\base\fontenc.sty"
Package: fontenc 2005/09/27 v1.99g Standard LaTeX package
("C:\Program Files\MiKTeX 2.9\tex\latex\base\t1enc.def"
File: t1enc.def 2005/09/27 v1.99g Standard LaTeX file
LaTeX Font Info: Redeclaring font encoding T1 on input line 43.
))
("C:\Program Files\MiKTeX 2.9\tex\latex\base\textcomp.sty"
Package: textcomp 2005/09/27 v1.99g Standard LaTeX package
Package textcomp Info: Sub-encoding information:
(textcomp) 5 = only ISO-Adobe without \textcurrency
(textcomp) 4 = 5 + \texteuro
(textcomp) 3 = 4 + \textohm
(textcomp) 2 = 3 + \textestimated + \textcurrency
(textcomp) 1 = TS1 - \textcircled - \t
(textcomp) 0 = TS1 (full)
(textcomp) Font families with sub-encoding setting implement
(textcomp) only a restricted character set as indicated.
(textcomp) Family '?' is the default used for unknown fonts.
(textcomp) See the documentation for details.
Package textcomp Info: Setting ? sub-encoding to TS1/1 on input line 71.
("C:\Program Files\MiKTeX 2.9\tex\latex\base\ts1enc.def"
File: ts1enc.def 2001/06/05 v3.0e (jk/car/fm) Standard LaTeX file
)
LaTeX Info: Redefining \oldstylenums on input line 266.
Package textcomp Info: Setting cmr sub-encoding to TS1/0 on input line 281.
Package textcomp Info: Setting cmss sub-encoding to TS1/0 on input line 282.
Package textcomp Info: Setting cmtt sub-encoding to TS1/0 on input line 283.
Package textcomp Info: Setting cmvtt sub-encoding to TS1/0 on input line 284.
Package textcomp Info: Setting cmbr sub-encoding to TS1/0 on input line 285.
Package textcomp Info: Setting cmtl sub-encoding to TS1/0 on input line 286.
Package textcomp Info: Setting ccr sub-encoding to TS1/0 on input line 287.
Package textcomp Info: Setting ptm sub-encoding to TS1/4 on input line 288.
Package textcomp Info: Setting pcr sub-encoding to TS1/4 on input line 289.
Package textcomp Info: Setting phv sub-encoding to TS1/4 on input line 290.
Package textcomp Info: Setting ppl sub-encoding to TS1/3 on input line 291.
Package textcomp Info: Setting pag sub-encoding to TS1/4 on input line 292.
Package textcomp Info: Setting pbk sub-encoding to TS1/4 on input line 293.
Package textcomp Info: Setting pnc sub-encoding to TS1/4 on input line 294.
Package textcomp Info: Setting pzc sub-encoding to TS1/4 on input line 295.
Package textcomp Info: Setting bch sub-encoding to TS1/4 on input line 296.
Package textcomp Info: Setting put sub-encoding to TS1/5 on input line 297.
Package textcomp Info: Setting uag sub-encoding to TS1/5 on input line 298.
Package textcomp Info: Setting ugq sub-encoding to TS1/5 on input line 299.
Package textcomp Info: Setting ul8 sub-encoding to TS1/4 on input line 300.
Package textcomp Info: Setting ul9 sub-encoding to TS1/4 on input line 301.
Package textcomp Info: Setting augie sub-encoding to TS1/5 on input line 302.
Package textcomp Info: Setting dayrom sub-encoding to TS1/3 on input line 303.
Package textcomp Info: Setting dayroms sub-encoding to TS1/3 on input line 304.
Package textcomp Info: Setting pxr sub-encoding to TS1/0 on input line 305.
Package textcomp Info: Setting pxss sub-encoding to TS1/0 on input line 306.
Package textcomp Info: Setting pxtt sub-encoding to TS1/0 on input line 307.
Package textcomp Info: Setting txr sub-encoding to TS1/0 on input line 308.
Package textcomp Info: Setting txss sub-encoding to TS1/0 on input line 309.
Package textcomp Info: Setting txtt sub-encoding to TS1/0 on input line 310.
Package textcomp Info: Setting lmr sub-encoding to TS1/0 on input line 311.
Package textcomp Info: Setting lmdh sub-encoding to TS1/0 on input line 312.
Package textcomp Info: Setting lmss sub-encoding to TS1/0 on input line 313.
Package textcomp Info: Setting lmssq sub-encoding to TS1/0 on input line 314.
Package textcomp Info: Setting lmvtt sub-encoding to TS1/0 on input line 315.
Package textcomp Info: Setting qhv sub-encoding to TS1/0 on input line 316.
Package textcomp Info: Setting qag sub-encoding to TS1/0 on input line 317.
Package textcomp Info: Setting qbk sub-encoding to TS1/0 on input line 318.
Package textcomp Info: Setting qcr sub-encoding to TS1/0 on input line 319.
Package textcomp Info: Setting qcs sub-encoding to TS1/0 on input line 320.
Package textcomp Info: Setting qpl sub-encoding to TS1/0 on input line 321.
Package textcomp Info: Setting qtm sub-encoding to TS1/0 on input line 322.
Package textcomp Info: Setting qzc sub-encoding to TS1/0 on input line 323.
Package textcomp Info: Setting qhvc sub-encoding to TS1/0 on input line 324.
Package textcomp Info: Setting futs sub-encoding to TS1/4 on input line 325.
Package textcomp Info: Setting futx sub-encoding to TS1/4 on input line 326.
Package textcomp Info: Setting futj sub-encoding to TS1/4 on input line 327.
Package textcomp Info: Setting hlh sub-encoding to TS1/3 on input line 328.
Package textcomp Info: Setting hls sub-encoding to TS1/3 on input line 329.
Package textcomp Info: Setting hlst sub-encoding to TS1/3 on input line 330.
Package textcomp Info: Setting hlct sub-encoding to TS1/5 on input line 331.
Package textcomp Info: Setting hlx sub-encoding to TS1/5 on input line 332.
Package textcomp Info: Setting hlce sub-encoding to TS1/5 on input line 333.
Package textcomp Info: Setting hlcn sub-encoding to TS1/5 on input line 334.
Package textcomp Info: Setting hlcw sub-encoding to TS1/5 on input line 335.
Package textcomp Info: Setting hlcf sub-encoding to TS1/5 on input line 336.
Package textcomp Info: Setting pplx sub-encoding to TS1/3 on input line 337.
Package textcomp Info: Setting pplj sub-encoding to TS1/3 on input line 338.
Package textcomp Info: Setting ptmx sub-encoding to TS1/4 on input line 339.
Package textcomp Info: Setting ptmj sub-encoding to TS1/4 on input line 340.
)
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\hyperref.sty"
Package: hyperref 2012/11/06 v6.83m Hypertext links for LaTeX
("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\hobsub-hyperref.sty"
Package: hobsub-hyperref 2012/04/25 v1.12 Bundle oberdiek, subset hyperref (HO)
("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\hobsub-generic.sty"
Package: hobsub-generic 2012/04/25 v1.12 Bundle oberdiek, subset generic (HO)
Package: hobsub 2012/04/25 v1.12 Construct package bundles (HO)
Package: infwarerr 2010/04/08 v1.3 Providing info/warning/error messages (HO)
Package: ltxcmds 2011/11/09 v1.22 LaTeX kernel commands for general use (HO)
Package: ifluatex 2010/03/01 v1.3 Provides the ifluatex switch (HO)
Package ifluatex Info: LuaTeX not detected.
Package: ifvtex 2010/03/01 v1.5 Detect VTeX and its facilities (HO)
Package ifvtex Info: VTeX not detected.
Package: intcalc 2007/09/27 v1.1 Expandable calculations with integers (HO)
Package: ifpdf 2011/01/30 v2.3 Provides the ifpdf switch (HO)
Package ifpdf Info: pdfTeX in PDF mode is not detected.
Package: etexcmds 2011/02/16 v1.5 Avoid name clashes with e-TeX commands (HO)
Package etexcmds Info: Could not find \expanded.
(etexcmds) That can mean that you are not using pdfTeX 1.50 or
(etexcmds) that some package has redefined \expanded.
(etexcmds) In the latter case, load this package earlier.
Package: kvsetkeys 2012/04/25 v1.16 Key value parser (HO)
Package: kvdefinekeys 2011/04/07 v1.3 Define keys (HO)
Package: pdftexcmds 2011/11/29 v0.20 Utility functions of pdfTeX for LuaTeX (HO
)
Package pdftexcmds Info: LuaTeX not detected.
Package pdftexcmds Info: \pdf@primitive is available.
Package pdftexcmds Info: \pdf@ifprimitive is available.
Package pdftexcmds Info: \pdfdraftmode is ignored in DVI mode.
Package: pdfescape 2011/11/25 v1.13 Implements pdfTeX's escape features (HO)
Package: bigintcalc 2012/04/08 v1.3 Expandable calculations on big integers (HO
)
Package: bitset 2011/01/30 v1.1 Handle bit-vector datatype (HO)
Package: uniquecounter 2011/01/30 v1.2 Provide unlimited unique counter (HO)
)
Package hobsub Info: Skipping package `hobsub' (already loaded).
Package: letltxmacro 2010/09/02 v1.4 Let assignment for LaTeX macros (HO)
Package: hopatch 2011/06/24 v1.1 Wrapper for package hooks (HO)
Package: xcolor-patch 2011/01/30 xcolor patch
Package: atveryend 2011/06/30 v1.8 Hooks at the very end of document (HO)
Package atveryend Info: \enddocument detected (standard20110627).
Package: atbegshi 2011/10/05 v1.16 At begin shipout hook (HO)
Package: refcount 2011/10/16 v3.4 Data extraction from label references (HO)
Package: hycolor 2011/01/30 v1.7 Color options for hyperref/bookmark (HO)
)
("C:\Program Files\MiKTeX 2.9\tex\generic\ifxetex\ifxetex.sty"
Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional
)
("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\auxhook.sty"
Package: auxhook 2011/03/04 v1.3 Hooks for auxiliary files (HO)
)
("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\kvoptions.sty"
Package: kvoptions 2011/06/30 v3.11 Key value format for package options (HO)
)
\@linkdim=\dimen103
\Hy@linkcounter=\count91
\Hy@pagecounter=\count92
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\pd1enc.def"
File: pd1enc.def 2012/11/06 v6.83m Hyperref: PDFDocEncoding definition (HO)
)
\Hy@SavedSpaceFactor=\count93
("C:\Program Files\MiKTeX 2.9\tex\latex\00miktex\hyperref.cfg"
File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
)
Package hyperref Warning: Unexpected value for option `pdftex'
(hyperref) is ignored on input line 4319.
! Package hyperref Error: Wrong driver option `pdftex',
(hyperref) because pdfTeX in PDF mode is not detected.
See the hyperref package documentation for explanation.
Type H <return> for immediate help.
...
l.4319 \ProcessKeyvalOptions{Hyp}
?
Package hyperref Info: Option `bookmarks' set `true' on input line 4319.
Package hyperref Warning: Option `a4paper' is no longer used.
Package hyperref Info: Option `colorlinks' set `true' on input line 4319.
Package hyperref Info: Option `breaklinks' set `true' on input line 4319.
Package hyperref Info: Hyper figures OFF on input line 4443.
Package hyperref Info: Link nesting OFF on input line 4448.
Package hyperref Info: Hyper index ON on input line 4451.
Package hyperref Info: Plain pages OFF on input line 4458.
Package hyperref Info: Backreferencing ON on input line 4461.
Package hyperref Info: Implicit mode ON; LaTeX internals redefined.
Package hyperref Info: Bookmarks ON on input line 4688.
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\backref.sty"
Package: backref 2012/07/25 v1.38 Bibliographical back referencing
("C:\Program Files\MiKTeX 2.9\tex\latex\oberdiek\rerunfilecheck.sty"
Package: rerunfilecheck 2011/04/15 v1.7 Rerun checks for auxiliary files (HO)
Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2
82.
))
\c@Hy@tempcnt=\count94
("C:\Program Files\MiKTeX 2.9\tex\latex\url\url.sty"
\Urlmuskip=\muskip10
Package: url 2013/09/16 ver 3.4 Verb mode for urls, etc.
)
LaTeX Info: Redefining \url on input line 5041.
\XeTeXLinkMargin=\dimen104
\Fld@menulength=\count95
\Field@Width=\dimen105
\Fld@charsize=\dimen106
Package hyperref Info: Hyper figures OFF on input line 6295.
Package hyperref Info: Link nesting OFF on input line 6300.
Package hyperref Info: Hyper index ON on input line 6303.
Package hyperref Info: backreferencing ON on input line 6308.
Package hyperref Info: Link coloring ON on input line 6313.
Package hyperref Info: Link coloring with OCG OFF on input line 6320.
Package hyperref Info: PDF/A mode OFF on input line 6325.
LaTeX Info: Redefining \ref on input line 6365.
LaTeX Info: Redefining \pageref on input line 6369.
\Hy@abspage=\count96
\c@Item=\count97
\c@Hfootnote=\count98
)
Package hyperref Message: Driver (default): hdvips.
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\hdvips.def"
File: hdvips.def 2012/11/06 v6.83m Hyperref driver for dvips
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\pdfmark.def"
File: pdfmark.def 2012/11/06 v6.83m Hyperref definitions for pdfmark specials
Package hyperref Warning: You have enabled option `breaklinks'.
(hyperref) But driver `hdvips.def' does not suppport this.
(hyperref) Expect trouble with the link areas of broken links.
\pdf@docset=\toks18
\pdf@box=\box27
\pdf@toks=\toks19
\pdf@defaulttoks=\toks20
\HyField@AnnotCount=\count99
\Fld@listcount=\count100
\c@bookmark@seq@number=\count101
\Hy@SectionHShift=\skip43
))
\pagenrlog=\write5
("C:\Program Files\MiKTeX 2.9\tex\latex\enumitem\enumitem.sty"
Package: enumitem 2011/09/28 v3.5.2 Customized lists
\labelindent=\skip44
\enit@outerparindent=\dimen107
\enit@toks=\toks21
\enit@inbox=\box28
\enitdp@description=\count102
)
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.aux
LaTeX Warning: Label `LeftDivision' multiply defined.
LaTeX Warning: Label `LeftDivision' multiply defined.
LaTeX Warning: Label `RightDivision' multiply defined.
LaTeX Warning: Label `RightDivision' multiply defined.
LaTeX Warning: Label `Sec:AutomorphicLoops' multiply defined.
)
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 87.
LaTeX Font Info: Try loading font information for TS1+cmr on input line 87.
("C:\Program Files\MiKTeX 2.9\tex\latex\base\ts1cmr.fd"
File: ts1cmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions
)
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Checking defaults for PD1/pdf/m/n on input line 87.
LaTeX Font Info: ... okay on input line 87.
LaTeX Font Info: Try loading font information for T1+ptm on input line 87.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\t1ptm.fd"
File: t1ptm.fd 2001/06/04 font definitions for T1/ptm.
)
\big@size=\dimen108
\AtBeginShipoutBox=\box29
Package hyperref Info: Link coloring ON on input line 87.
("C:\Program Files\MiKTeX 2.9\tex\latex\hyperref\nameref.sty"
Package: nameref 2012/10/27 v2.43 Cross-referencing by name of section
("C:\Program Files\MiKTeX 2.9\tex\generic\oberdiek\gettitlestring.sty"
Package: gettitlestring 2010/12/03 v1.4 Cleanup title references (HO)
)
\c@section@level=\count103
)
LaTeX Info: Redefining \ref on input line 87.
LaTeX Info: Redefining \pageref on input line 87.
LaTeX Info: Redefining \nameref on input line 87.
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.out)
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.out)
\@outlinefile=\write6
Package hyperref Warning: Rerun to get /PageLabels entry.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <50> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 93.
LaTeX Font Info: Try loading font information for T1+phv on input line 93.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\t1phv.fd"
File: t1phv.fd 2001/06/04 scalable font definitions for T1/phv.
)
LaTeX Font Info: Font shape `T1/phv/bx/n' in size <50> not available
(Font) Font shape `T1/phv/b/n' tried instead on input line 93.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <24.88> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 98.
LaTeX Font Info: Font shape `T1/phv/bx/n' in size <24.88> not available
(Font) Font shape `T1/phv/b/n' tried instead on input line 98.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <14.4> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 103.
Package hyperref Warning: Composite letter `\textasciicaron+e'
(hyperref) not defined in PD1 encoding,
(hyperref) removing `\textasciicaron' on input line 105.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <10> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 110.
LaTeX Font Info: Try loading font information for T1+cmtt on input line 110.
("C:\Program Files\MiKTeX 2.9\tex\latex\base\t1cmtt.fd"
File: t1cmtt.fd 1999/05/25 v2.5h Standard LaTeX font definitions
)
Underfull \hbox (badness 10000) in paragraph at lines 108--119
[]
[1
]
LaTeX Font Info: Try loading font information for TS1+ptm on input line 125.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\ts1ptm.fd"
File: ts1ptm.fd 2001/06/04 font definitions for TS1/ptm.
)
Underfull \hbox (badness 10000) in paragraph at lines 125--126
[]
[2] (C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.toc
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <10.95> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 1.
LaTeX Font Info: Try loading font information for OT1+ztmcm on input line 2.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\ot1ztmcm.fd"
File: ot1ztmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OT1/ztmcm.
)
LaTeX Font Info: Try loading font information for OML+ztmcm on input line 2.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\omlztmcm.fd"
File: omlztmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OML/ztmcm.
)
LaTeX Font Info: Try loading font information for OMS+ztmcm on input line 2.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\omsztmcm.fd"
File: omsztmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OMS/ztmcm.
)
LaTeX Font Info: Try loading font information for OMX+ztmcm on input line 2.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\omxztmcm.fd"
File: omxztmcm.fd 2000/01/03 Fontinst v1.801 font definitions for OMX/ztmcm.
)
LaTeX Font Info: Try loading font information for OT1+ptm on input line 2.
("C:\Program Files\MiKTeX 2.9\tex\latex\psnfss\ot1ptm.fd"
File: ot1ptm.fd 2001/06/04 font definitions for OT1/ptm.
)
LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <10.95> not available
(Font) Font shape `OT1/ptm/b/n' tried instead on input line 2.
LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <8> not available
(Font) Font shape `OT1/ptm/b/n' tried instead on input line 2.
LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <6> not available
(Font) Font shape `OT1/ptm/b/n' tried instead on input line 2.
[3
] [4])
\tf@toc=\write7
[5]
Chapter 1.
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <20.74> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 134.
[6
] [7]
Chapter 2.
[8
] [9] [10]
Chapter 3.
[11
]
Underfull \hbox (badness 1264) in paragraph at lines 394--396
[][]\T1/ptm/m/n/10.95 In some spe-cial sit-u-a-tions, the func-tion \T1/cmtt/m/
n/10.95 IntoSomething([]) \T1/ptm/m/n/10.95 al-lows to con-vert [] into
[]
Underfull \hbox (badness 10000) in paragraph at lines 404--408
[]
Underfull \hbox (badness 10000) in paragraph at lines 410--413
[]
Underfull \hbox (badness 10000) in paragraph at lines 415--417
[]
Underfull \hbox (badness 10000) in paragraph at lines 419--423
[]
[12]
LaTeX Font Info: Font shape `T1/ptm/bx/n' in size <12> not available
(Font) Font shape `T1/ptm/b/n' tried instead on input line 451.
Underfull \hbox (badness 10000) in paragraph at lines 454--460
[]
Underfull \hbox (badness 10000) in paragraph at lines 464--466
[]
Underfull \hbox (badness 10000) in paragraph at lines 469--471
[]
LaTeX Font Info: Font shape `T1/cmtt/bx/n' in size <10> not available
(Font) Font shape `T1/cmtt/m/n' tried instead on input line 476.
[13]
Chapter 4.
Underfull \hbox (badness 10000) in paragraph at lines 538--546
[]
[14
] [15]
Underfull \hbox (badness 10000) in paragraph at lines 669--671
[]
[16]
Underfull \hbox (badness 10000) in paragraph at lines 763--769
[]
[17]
Underfull \hbox (badness 10000) in paragraph at lines 813--815
[]
[18] [19]
Underfull \hbox (badness 10000) in paragraph at lines 945--947
[]
Underfull \hbox (badness 10000) in paragraph at lines 984--990
[]
Underfull \hbox (badness 2401) in paragraph at lines 1001--1007
[]\T1/ptm/b/n/10.95 Returns: \T1/ptm/m/n/10.95 If [] is a de-clared magma that
hap-pens to be a group, the cor-re-spond-
[]
[20] [21]
Chapter 5.
[22
]
Underfull \hbox (badness 10000) in paragraph at lines 1165--1173
[]
Underfull \hbox (badness 10000) in paragraph at lines 1175--1190
[]
[23] [24] [25]
Chapter 6.
Underfull \hbox (badness 10000) in paragraph at lines 1378--1383
[]
Underfull \hbox (badness 10000) in paragraph at lines 1401--1407
[]
[26
]
Underfull \hbox (badness 10000) in paragraph at lines 1446--1449
[]
[27]
Underfull \hbox (badness 10000) in paragraph at lines 1554--1562
[]
[28] [29]
Underfull \hbox (badness 1221) in paragraph at lines 1685--1685
[][]\T1/ptm/b/n/12 LeftInnerMappingGroup, Right-In-nerMap-ping-Group, Mid-dleIn
-nerMap-ping-
[]
Underfull \hbox (badness 10000) in paragraph at lines 1708--1713
[]
[30] [31] [32]
Underfull \hbox (badness 10000) in paragraph at lines 2068--2072
[]
[33]
Underfull \hbox (badness 10000) in paragraph at lines 2139--2144
[]
Underfull \hbox (badness 10000) in paragraph at lines 2150--2151
[]
Underfull \hbox (badness 10000) in paragraph at lines 2173--2178
[]
[34] [35]
Chapter 7.
[36
]
Underfull \hbox (badness 10000) in paragraph at lines 2419--2421
[]
[37] [38] [39] [40]
Underfull \hbox (badness 10000) in paragraph at lines 2769--2774
[]
Underfull \hbox (badness 10000) in paragraph at lines 2780--2782
[]
[41] [42] [43] [44]
Chapter 8.
[45
] [46]
Underfull \hbox (badness 10000) in paragraph at lines 3224--3227
[]
[47] [48]
Chapter 9.
Underfull \hbox (badness 10000) in paragraph at lines 3361--3362
[][]\T1/cmtt/m/n/10.95 LOOPS_my_library_data[2][k] \T1/ptm/m/n/10.95 is the num
-ber of loops of or-der
[]
Underfull \hbox (badness 10000) in paragraph at lines 3383--3386
[]
Underfull \hbox (badness 10000) in paragraph at lines 3397--3402
[]
[49
] [50] [51]
Underfull \hbox (badness 10000) in paragraph at lines 3588--3591
[]
[52] [53] [54]
Appendix A.
Underfull \hbox (badness 10000) in paragraph at lines 3788--3833
[]
[55
] [56]
Appendix B.
Underfull \hbox (badness 10000) in paragraph at lines 3842--3962
[]
Underfull \hbox (badness 10000) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( HasAntiautomorphicInverseProperty, HasAutomorphicInversePr
operty and
[]
Underfull \hbox (badness 10000) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( HasAutomorphicInverseProperty, HasAntiautomorphicInversePr
operty and
[]
Overfull \hbox (21.33267pt too wide) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( HasInverseProperty, HasLeftInverseProperty and HasAntiauto
morphicInverseProperty
[]
Overfull \hbox (26.99104pt too wide) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( HasInverseProperty, HasRightInverseProperty and HasAntiaut
omorphicInverseProperty
[]
Overfull \hbox (21.33267pt too wide) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( HasInverseProperty, HasWeakInverseProperty and HasAntiauto
morphicInverseProperty
[]
Overfull \hbox (38.30779pt too wide) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntia
utomorphicInverseProperty
[]
Overfull \hbox (38.30779pt too wide) in paragraph at lines 3842--3962
\T1/cmtt/m/n/10.95 ( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntia
utomorphicInverseProperty
[]
[57
] [58] (C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.bbl
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.brf)
\tf@brf=\write8
[59] [60
]) [61]
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.ind [62
] [63] [64] [65]
Underfull \hbox (badness 10000) in paragraph at lines 430--432
[][]\T1/cmtt/m/n/10.95 Right\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Bol\T1/ptm/m/
n/10.95 -\T1/cmtt/m/n/10.95 Loop\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 By\T1/ptm
/m/n/10.95 -\T1/cmtt/m/n/10.95 Exact\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Group
\T1/ptm/m/n/10.95 -\T1/cmtt/m/n/10.95 Factorization\T1/ptm/m/n/10.95 ,
[]
[66])
Package atveryend Info: Empty hook `BeforeClearDocument' on input line 3986.
Package atveryend Info: Empty hook `AfterLastShipout' on input line 3986.
(C:\cygwin64\opt\gap4r7\pkg\loops\doc\loops.aux)
Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 3986.
Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 3986.
Package rerunfilecheck Info: File `loops.out' has not changed.
(rerunfilecheck) Checksum: B35AC9998206B5E4DDDBB0BB03F0A272;5893.
Package rerunfilecheck Info: File `loops.brf' has not changed.
(rerunfilecheck) Checksum: C248D12B2C67B5CBC4D9D8FC93C1CB81;1554.
LaTeX Warning: There were multiply-defined labels.
Package atveryend Info: Empty hook `AtVeryVeryEnd' on input line 3986.
)
Here is how much of TeX's memory you used:
7134 strings out of 493922
107334 string characters out of 3147269
214990 words of memory out of 3000000
10330 multiletter control sequences out of 15000+200000
56657 words of font info for 68 fonts, out of 3000000 for 9000
841 hyphenation exceptions out of 8191
32i,7n,29p,918b,613s stack positions out of 5000i,500n,10000p,200000b,50000s
Output written on loops.dvi (66 pages, 548296 bytes).

83
doc/loops.out
View File

@ -1,83 +0,0 @@
\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
\BOOKMARK [1][-]{section.1.1}{License}{chapter.1}% 2
\BOOKMARK [1][-]{section.1.2}{Installation}{chapter.1}% 3
\BOOKMARK [1][-]{section.1.3}{Documentation}{chapter.1}% 4
\BOOKMARK [1][-]{section.1.4}{Test Files}{chapter.1}% 5
\BOOKMARK [1][-]{section.1.5}{Memory Management}{chapter.1}% 6
\BOOKMARK [1][-]{section.1.6}{Feedback}{chapter.1}% 7
\BOOKMARK [1][-]{section.1.7}{Acknowledgment}{chapter.1}% 8
\BOOKMARK [0][-]{chapter.2}{Mathematical Background}{}% 9
\BOOKMARK [1][-]{section.2.1}{Quasigroups and Loops}{chapter.2}% 10
\BOOKMARK [1][-]{section.2.2}{Translations}{chapter.2}% 11
\BOOKMARK [1][-]{section.2.3}{Subquasigroups and Subloops}{chapter.2}% 12
\BOOKMARK [1][-]{section.2.4}{Nilpotence and Solvability}{chapter.2}% 13
\BOOKMARK [1][-]{section.2.5}{Associators and Commutators}{chapter.2}% 14
\BOOKMARK [1][-]{section.2.6}{Homomorphism and Homotopisms}{chapter.2}% 15
\BOOKMARK [0][-]{chapter.3}{How the Package Works}{}% 16
\BOOKMARK [1][-]{section.3.1}{Representing Quasigroups}{chapter.3}% 17
\BOOKMARK [1][-]{section.3.2}{Conversions between magmas, quasigroups, loops and groups}{chapter.3}% 18
\BOOKMARK [1][-]{section.3.3}{Calculating with Quasigroups}{chapter.3}% 19
\BOOKMARK [1][-]{section.3.4}{Naming, Viewing and Printing Quasigroups and their Elements}{chapter.3}% 20
\BOOKMARK [0][-]{chapter.4}{Creating Quasigroups and Loops}{}% 21
\BOOKMARK [1][-]{section.4.1}{About Cayley Tables}{chapter.4}% 22
\BOOKMARK [1][-]{section.4.2}{Testing Cayley Tables}{chapter.4}% 23
\BOOKMARK [1][-]{section.4.3}{Canonical and Normalized Cayley Tables}{chapter.4}% 24
\BOOKMARK [1][-]{section.4.4}{Creating Quasigroups and Loops From Cayley Tables}{chapter.4}% 25
\BOOKMARK [1][-]{section.4.5}{Creating Quasigroups and Loops from a File}{chapter.4}% 26
\BOOKMARK [1][-]{section.4.6}{Creating Quasigroups and Loops From Sections}{chapter.4}% 27
\BOOKMARK [1][-]{section.4.7}{Creating Quasigroups and Loops From Folders}{chapter.4}% 28
\BOOKMARK [1][-]{section.4.8}{Creating Quasigroups and Loops By Nuclear Extensions}{chapter.4}% 29
\BOOKMARK [1][-]{section.4.9}{Random Quasigroups and Loops}{chapter.4}% 30
\BOOKMARK [1][-]{section.4.10}{Conversions}{chapter.4}% 31
\BOOKMARK [1][-]{section.4.11}{Products of Quasigroups and Loops}{chapter.4}% 32
\BOOKMARK [1][-]{section.4.12}{Opposite Quasigroups and Loops}{chapter.4}% 33
\BOOKMARK [0][-]{chapter.5}{Basic Methods And Attributes}{}% 34
\BOOKMARK [1][-]{section.5.1}{Basic Attributes}{chapter.5}% 35
\BOOKMARK [1][-]{section.5.2}{Basic Arithmetic Operations}{chapter.5}% 36
\BOOKMARK [1][-]{section.5.3}{Powers and Inverses}{chapter.5}% 37
\BOOKMARK [1][-]{section.5.4}{Associators and Commutators}{chapter.5}% 38
\BOOKMARK [1][-]{section.5.5}{Generators}{chapter.5}% 39
\BOOKMARK [0][-]{chapter.6}{Methods Based on Permutation Groups}{}% 40
\BOOKMARK [1][-]{section.6.1}{Parent of a Quasigroup}{chapter.6}% 41
\BOOKMARK [1][-]{section.6.2}{Subquasigroups and Subloops}{chapter.6}% 42
\BOOKMARK [1][-]{section.6.3}{Translations and Sections}{chapter.6}% 43
\BOOKMARK [1][-]{section.6.4}{Multiplication Groups}{chapter.6}% 44
\BOOKMARK [1][-]{section.6.5}{Inner Mapping Groups}{chapter.6}% 45
\BOOKMARK [1][-]{section.6.6}{Nuclei, Commutant, Center, and Associator Subloop}{chapter.6}% 46
\BOOKMARK [1][-]{section.6.7}{Normal Subloops and Simple Loops}{chapter.6}% 47
\BOOKMARK [1][-]{section.6.8}{Factor Loops}{chapter.6}% 48
\BOOKMARK [1][-]{section.6.9}{Nilpotency and Central Series}{chapter.6}% 49
\BOOKMARK [1][-]{section.6.10}{Solvability, Derived Series and Frattini Subloop}{chapter.6}% 50
\BOOKMARK [1][-]{section.6.11}{Isomorphisms and Automorphisms}{chapter.6}% 51
\BOOKMARK [1][-]{section.6.12}{Isotopisms}{chapter.6}% 52
\BOOKMARK [0][-]{chapter.7}{Testing Properties of Quasigroups and Loops}{}% 53
\BOOKMARK [1][-]{section.7.1}{Associativity, Commutativity and Generalizations}{chapter.7}% 54
\BOOKMARK [1][-]{section.7.2}{Inverse Propeties}{chapter.7}% 55
\BOOKMARK [1][-]{section.7.3}{Some Properties of Quasigroups}{chapter.7}% 56
\BOOKMARK [1][-]{section.7.4}{Loops of Bol Moufang Type}{chapter.7}% 57
\BOOKMARK [1][-]{section.7.5}{Power Alternative Loops}{chapter.7}% 58
\BOOKMARK [1][-]{section.7.6}{Conjugacy Closed Loops and Related Properties}{chapter.7}% 59
\BOOKMARK [1][-]{section.7.7}{Automorphic Loops}{chapter.7}% 60
\BOOKMARK [1][-]{section.7.8}{Additonal Varieties of Loops}{chapter.7}% 61
\BOOKMARK [0][-]{chapter.8}{Specific Methods}{}% 62
\BOOKMARK [1][-]{section.8.1}{Core Methods for Bol Loops}{chapter.8}% 63
\BOOKMARK [1][-]{section.8.2}{Moufang Modifications}{chapter.8}% 64
\BOOKMARK [1][-]{section.8.3}{Triality for Moufang Loops}{chapter.8}% 65
\BOOKMARK [1][-]{section.8.4}{Realizing Groups as Multiplication Groups of Loops}{chapter.8}% 66
\BOOKMARK [0][-]{chapter.9}{Libraries of Loops}{}% 67
\BOOKMARK [1][-]{section.9.1}{A Typical Library}{chapter.9}% 68
\BOOKMARK [1][-]{section.9.2}{Left Bol Loops and Right Bol Loops}{chapter.9}% 69
\BOOKMARK [1][-]{section.9.3}{Moufang Loops}{chapter.9}% 70
\BOOKMARK [1][-]{section.9.4}{Code Loops}{chapter.9}% 71
\BOOKMARK [1][-]{section.9.5}{Steiner Loops}{chapter.9}% 72
\BOOKMARK [1][-]{section.9.6}{Conjugacy Closed Loops}{chapter.9}% 73
\BOOKMARK [1][-]{section.9.7}{Small Loops}{chapter.9}% 74
\BOOKMARK [1][-]{section.9.8}{Paige Loops}{chapter.9}% 75
\BOOKMARK [1][-]{section.9.9}{Nilpotent Loops}{chapter.9}% 76
\BOOKMARK [1][-]{section.9.10}{Automorphic Loops}{chapter.9}% 77
\BOOKMARK [1][-]{section.9.11}{Interesting Loops}{chapter.9}% 78
\BOOKMARK [1][-]{section.9.12}{Libraries of Loops Up To Isotopism}{chapter.9}% 79
\BOOKMARK [0][-]{appendix.A}{Files}{}% 80
\BOOKMARK [0][-]{appendix.B}{Filters}{}% 81
\BOOKMARK [0][-]{appendix*.3}{References}{}% 82
\BOOKMARK [0][-]{section*.4}{Index}{}% 83

246
doc/loops.pnr
View File

@ -1,246 +0,0 @@
PAGENRS := [
[ 0, 0, 0 ], 1,
[ 0, 0, 1 ], 2,
[ 0, 0, 2 ], 3,
[ 1, 0, 0 ], 6,
[ 1, 1, 0 ], 6,
[ 1, 2, 0 ], 6,
[ 1, 3, 0 ], 6,
[ 1, 4, 0 ], 7,
[ 1, 5, 0 ], 7,
[ 1, 6, 0 ], 7,
[ 1, 7, 0 ], 7,
[ 2, 0, 0 ], 8,
[ 2, 1, 0 ], 8,
[ 2, 2, 0 ], 8,
[ 2, 3, 0 ], 9,
[ 2, 4, 0 ], 9,
[ 2, 5, 0 ], 9,
[ 2, 6, 0 ], 9,
[ 3, 0, 0 ], 11,
[ 3, 1, 0 ], 11,
[ 3, 2, 0 ], 12,
[ 3, 3, 0 ], 12,
[ 3, 4, 0 ], 13,
[ 3, 4, 1 ], 13,
[ 4, 0, 0 ], 14,
[ 4, 1, 0 ], 14,
[ 4, 2, 0 ], 14,
[ 4, 2, 1 ], 14,
[ 4, 2, 2 ], 14,
[ 4, 3, 0 ], 15,
[ 4, 3, 1 ], 15,
[ 4, 3, 2 ], 15,
[ 4, 3, 3 ], 15,
[ 4, 4, 0 ], 15,
[ 4, 4, 1 ], 15,
[ 4, 5, 0 ], 16,
[ 4, 5, 1 ], 17,
[ 4, 6, 0 ], 17,
[ 4, 6, 1 ], 17,
[ 4, 6, 2 ], 17,
[ 4, 6, 3 ], 17,
[ 4, 7, 0 ], 18,
[ 4, 7, 1 ], 18,
[ 4, 8, 0 ], 18,
[ 4, 8, 1 ], 18,
[ 4, 8, 2 ], 18,
[ 4, 9, 0 ], 19,
[ 4, 9, 1 ], 19,
[ 4, 9, 2 ], 19,
[ 4, 10, 0 ], 20,
[ 4, 10, 1 ], 20,
[ 4, 10, 2 ], 20,
[ 4, 10, 3 ], 20,
[ 4, 10, 4 ], 20,
[ 4, 11, 0 ], 21,
[ 4, 11, 1 ], 21,
[ 4, 12, 0 ], 21,
[ 4, 12, 1 ], 21,
[ 5, 0, 0 ], 22,
[ 5, 1, 0 ], 22,
[ 5, 1, 1 ], 22,
[ 5, 1, 2 ], 22,
[ 5, 1, 3 ], 22,
[ 5, 1, 4 ], 22,
[ 5, 1, 5 ], 23,
[ 5, 2, 0 ], 23,
[ 5, 2, 1 ], 23,
[ 5, 2, 2 ], 23,
[ 5, 3, 0 ], 23,
[ 5, 3, 1 ], 24,
[ 5, 4, 0 ], 24,
[ 5, 4, 1 ], 24,
[ 5, 4, 2 ], 24,
[ 5, 5, 0 ], 24,
[ 5, 5, 1 ], 24,
[ 5, 5, 2 ], 25,
[ 5, 5, 3 ], 25,
[ 6, 0, 0 ], 26,
[ 6, 1, 0 ], 26,
[ 6, 1, 1 ], 26,
[ 6, 1, 2 ], 26,
[ 6, 1, 3 ], 27,
[ 6, 2, 0 ], 27,
[ 6, 2, 1 ], 27,
[ 6, 2, 2 ], 27,
[ 6, 2, 3 ], 27,
[ 6, 2, 4 ], 27,
[ 6, 2, 5 ], 28,
[ 6, 2, 6 ], 28,
[ 6, 2, 7 ], 28,
[ 6, 3, 0 ], 28,
[ 6, 3, 1 ], 28,
[ 6, 3, 2 ], 28,
[ 6, 4, 0 ], 29,
[ 6, 4, 1 ], 29,
[ 6, 4, 2 ], 29,
[ 6, 5, 0 ], 29,
[ 6, 5, 1 ], 30,
[ 6, 5, 2 ], 30,
[ 6, 5, 3 ], 30,
[ 6, 6, 0 ], 30,
[ 6, 6, 1 ], 30,
[ 6, 6, 2 ], 31,
[ 6, 6, 3 ], 31,
[ 6, 6, 4 ], 31,
[ 6, 6, 5 ], 31,
[ 6, 7, 0 ], 31,
[ 6, 7, 1 ], 31,
[ 6, 7, 2 ], 31,
[ 6, 7, 3 ], 32,
[ 6, 8, 0 ], 32,
[ 6, 8, 1 ], 32,
[ 6, 8, 2 ], 32,
[ 6, 9, 0 ], 32,
[ 6, 9, 1 ], 32,
[ 6, 9, 2 ], 32,
[ 6, 9, 3 ], 32,
[ 6, 9, 4 ], 33,
[ 6, 9, 5 ], 33,
[ 6, 10, 0 ], 33,
[ 6, 10, 1 ], 33,
[ 6, 10, 2 ], 33,
[ 6, 10, 3 ], 33,
[ 6, 10, 4 ], 33,
[ 6, 10, 5 ], 33,
[ 6, 11, 0 ], 33,
[ 6, 11, 1 ], 33,
[ 6, 11, 2 ], 34,
[ 6, 11, 3 ], 34,
[ 6, 11, 4 ], 34,
[ 6, 11, 5 ], 34,
[ 6, 11, 6 ], 34,
[ 6, 11, 7 ], 34,
[ 6, 11, 8 ], 35,
[ 6, 11, 9 ], 35,
[ 6, 12, 0 ], 35,
[ 6, 12, 1 ], 35,
[ 6, 12, 2 ], 35,
[ 7, 0, 0 ], 36,
[ 7, 1, 0 ], 36,
[ 7, 1, 1 ], 36,
[ 7, 1, 2 ], 36,
[ 7, 1, 3 ], 36,
[ 7, 1, 4 ], 36,
[ 7, 2, 0 ], 37,
[ 7, 2, 1 ], 37,
[ 7, 2, 2 ], 37,
[ 7, 2, 3 ], 37,
[ 7, 2, 4 ], 37,
[ 7, 2, 5 ], 37,
[ 7, 3, 0 ], 38,
[ 7, 3, 1 ], 38,
[ 7, 3, 2 ], 38,
[ 7, 3, 3 ], 38,
[ 7, 3, 4 ], 38,
[ 7, 3, 5 ], 38,
[ 7, 3, 6 ], 38,
[ 7, 3, 7 ], 39,
[ 7, 4, 0 ], 39,
[ 7, 4, 1 ], 40,
[ 7, 4, 2 ], 40,
[ 7, 4, 3 ], 40,
[ 7, 4, 4 ], 40,
[ 7, 4, 5 ], 40,
[ 7, 4, 6 ], 40,
[ 7, 4, 7 ], 40,
[ 7, 4, 8 ], 40,
[ 7, 4, 9 ], 40,
[ 7, 4, 10 ], 40,
[ 7, 4, 11 ], 41,
[ 7, 4, 12 ], 41,
[ 7, 4, 13 ], 41,
[ 7, 4, 14 ], 41,
[ 7, 4, 15 ], 41,
[ 7, 5, 0 ], 42,
[ 7, 5, 1 ], 42,
[ 7, 6, 0 ], 42,
[ 7, 6, 1 ], 42,
[ 7, 6, 2 ], 42,
[ 7, 6, 3 ], 42,
[ 7, 6, 4 ], 42,
[ 7, 7, 0 ], 43,
[ 7, 7, 1 ], 43,
[ 7, 7, 2 ], 43,
[ 7, 7, 3 ], 44,
[ 7, 7, 4 ], 44,
[ 7, 8, 0 ], 44,
[ 7, 8, 1 ], 44,
[ 7, 8, 2 ], 44,
[ 7, 8, 3 ], 44,
[ 7, 8, 4 ], 44,
[ 8, 0, 0 ], 45,
[ 8, 1, 0 ], 45,
[ 8, 1, 1 ], 45,
[ 8, 1, 2 ], 45,
[ 8, 1, 3 ], 45,
[ 8, 2, 0 ], 46,
[ 8, 2, 1 ], 46,
[ 8, 2, 2 ], 46,
[ 8, 2, 3 ], 46,
[ 8, 3, 0 ], 46,
[ 8, 3, 1 ], 47,
[ 8, 3, 2 ], 47,
[ 8, 4, 0 ], 47,
[ 8, 4, 1 ], 47,
[ 8, 4, 2 ], 47,
[ 8, 4, 3 ], 47,
[ 8, 4, 4 ], 48,
[ 8, 4, 5 ], 48,
[ 8, 4, 6 ], 48,
[ 9, 0, 0 ], 49,
[ 9, 1, 0 ], 49,
[ 9, 1, 1 ], 49,
[ 9, 1, 2 ], 49,
[ 9, 1, 3 ], 50,
[ 9, 2, 0 ], 50,
[ 9, 2, 1 ], 50,
[ 9, 2, 2 ], 50,
[ 9, 3, 0 ], 50,
[ 9, 3, 1 ], 50,
[ 9, 4, 0 ], 51,
[ 9, 4, 1 ], 51,
[ 9, 5, 0 ], 51,
[ 9, 5, 1 ], 51,
[ 9, 6, 0 ], 51,
[ 9, 6, 1 ], 52,
[ 9, 6, 2 ], 52,
[ 9, 6, 3 ], 52,
[ 9, 7, 0 ], 52,
[ 9, 7, 1 ], 53,
[ 9, 8, 0 ], 53,
[ 9, 8, 1 ], 53,
[ 9, 9, 0 ], 53,
[ 9, 9, 1 ], 53,
[ 9, 10, 0 ], 53,
[ 9, 10, 1 ], 53,
[ 9, 11, 0 ], 54,
[ 9, 11, 1 ], 54,
[ 9, 12, 0 ], 54,
[ 9, 12, 1 ], 54,
[ "A", 0, 0 ], 55,
[ "B", 0, 0 ], 57,
[ "Bib", 0, 0 ], 60,
[ "Ind", 0, 0 ], 62,
["End"], 67];

3930
doc/loops.tex
View File

File diff suppressed because it is too large Load Diff

241
doc/loops.toc
View File

@ -1,241 +0,0 @@
\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter }Introduction}}{6}{chapter.1}
\contentsline {section}{\numberline {1.1}\leavevmode {\color {Chapter }License}}{6}{section.1.1}
\contentsline {section}{\numberline {1.2}\leavevmode {\color {Chapter }Installation}}{6}{section.1.2}
\contentsline {section}{\numberline {1.3}\leavevmode {\color {Chapter }Documentation}}{6}{section.1.3}
\contentsline {section}{\numberline {1.4}\leavevmode {\color {Chapter }Test Files}}{7}{section.1.4}
\contentsline {section}{\numberline {1.5}\leavevmode {\color {Chapter }Memory Management}}{7}{section.1.5}
\contentsline {section}{\numberline {1.6}\leavevmode {\color {Chapter }Feedback}}{7}{section.1.6}
\contentsline {section}{\numberline {1.7}\leavevmode {\color {Chapter }Acknowledgment}}{7}{section.1.7}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter }Mathematical Background}}{8}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter }Quasigroups and Loops}}{8}{section.2.1}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Translations}}{8}{section.2.2}
\contentsline {section}{\numberline {2.3}\leavevmode {\color {Chapter }Subquasigroups and Subloops}}{9}{section.2.3}
\contentsline {section}{\numberline {2.4}\leavevmode {\color {Chapter }Nilpotence and Solvability}}{9}{section.2.4}
\contentsline {section}{\numberline {2.5}\leavevmode {\color {Chapter }Associators and Commutators}}{9}{section.2.5}
\contentsline {section}{\numberline {2.6}\leavevmode {\color {Chapter }Homomorphism and Homotopisms}}{9}{section.2.6}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter }How the Package Works}}{11}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter }Representing Quasigroups}}{11}{section.3.1}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Conversions between magmas, quasigroups, loops and groups}}{12}{section.3.2}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Calculating with Quasigroups}}{12}{section.3.3}
\contentsline {section}{\numberline {3.4}\leavevmode {\color {Chapter }Naming, Viewing and Printing Quasigroups and their Elements}}{13}{section.3.4}
\contentsline {subsection}{\numberline {3.4.1}\leavevmode {\color {Chapter }SetQuasigroupElmName and SetLoopElmName}}{13}{subsection.3.4.1}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter }Creating Quasigroups and Loops}}{14}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }About Cayley Tables}}{14}{section.4.1}
\contentsline {section}{\numberline {4.2}\leavevmode {\color {Chapter }Testing Cayley Tables}}{14}{section.4.2}
\contentsline {subsection}{\numberline {4.2.1}\leavevmode {\color {Chapter }IsQuasigroupTable and IsQuasigroupCayleyTable}}{14}{subsection.4.2.1}
\contentsline {subsection}{\numberline {4.2.2}\leavevmode {\color {Chapter }IsLoopTable and IsLoopCayleyTable}}{14}{subsection.4.2.2}
\contentsline {section}{\numberline {4.3}\leavevmode {\color {Chapter }Canonical and Normalized Cayley Tables}}{15}{section.4.3}
\contentsline {subsection}{\numberline {4.3.1}\leavevmode {\color {Chapter }CanonicalCayleyTable}}{15}{subsection.4.3.1}
\contentsline {subsection}{\numberline {4.3.2}\leavevmode {\color {Chapter }CanonicalCopy}}{15}{subsection.4.3.2}
\contentsline {subsection}{\numberline {4.3.3}\leavevmode {\color {Chapter }NormalizedQuasigroupTable}}{15}{subsection.4.3.3}
\contentsline {section}{\numberline {4.4}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Cayley Tables}}{15}{section.4.4}
\contentsline {subsection}{\numberline {4.4.1}\leavevmode {\color {Chapter }QuasigroupByCayleyTable and LoopByCayleyTable}}{15}{subsection.4.4.1}
\contentsline {section}{\numberline {4.5}\leavevmode {\color {Chapter }Creating Quasigroups and Loops from a File}}{16}{section.4.5}
\contentsline {subsection}{\numberline {4.5.1}\leavevmode {\color {Chapter }QuasigroupFromFile and LoopFromFile}}{17}{subsection.4.5.1}
\contentsline {section}{\numberline {4.6}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Sections}}{17}{section.4.6}
\contentsline {subsection}{\numberline {4.6.1}\leavevmode {\color {Chapter }CayleyTableByPerms}}{17}{subsection.4.6.1}
\contentsline {subsection}{\numberline {4.6.2}\leavevmode {\color {Chapter }QuasigroupByLeftSection and LoopByLeftSection}}{17}{subsection.4.6.2}
\contentsline {subsection}{\numberline {4.6.3}\leavevmode {\color {Chapter }QuasigroupByRightSection and LoopByRightSection}}{17}{subsection.4.6.3}
\contentsline {section}{\numberline {4.7}\leavevmode {\color {Chapter }Creating Quasigroups and Loops From Folders}}{18}{section.4.7}
\contentsline {subsection}{\numberline {4.7.1}\leavevmode {\color {Chapter }QuasigroupByRightFolder and LoopByRightFolder}}{18}{subsection.4.7.1}
\contentsline {section}{\numberline {4.8}\leavevmode {\color {Chapter }Creating Quasigroups and Loops By Nuclear Extensions}}{18}{section.4.8}
\contentsline {subsection}{\numberline {4.8.1}\leavevmode {\color {Chapter }NuclearExtension}}{18}{subsection.4.8.1}
\contentsline {subsection}{\numberline {4.8.2}\leavevmode {\color {Chapter }LoopByExtension}}{18}{subsection.4.8.2}
\contentsline {section}{\numberline {4.9}\leavevmode {\color {Chapter }Random Quasigroups and Loops}}{19}{section.4.9}
\contentsline {subsection}{\numberline {4.9.1}\leavevmode {\color {Chapter }RandomQuasigroup and RandomLoop}}{19}{subsection.4.9.1}
\contentsline {subsection}{\numberline {4.9.2}\leavevmode {\color {Chapter }RandomNilpotentLoop}}{19}{subsection.4.9.2}
\contentsline {section}{\numberline {4.10}\leavevmode {\color {Chapter }Conversions}}{20}{section.4.10}
\contentsline {subsection}{\numberline {4.10.1}\leavevmode {\color {Chapter }IntoQuasigroup}}{20}{subsection.4.10.1}
\contentsline {subsection}{\numberline {4.10.2}\leavevmode {\color {Chapter }PrincipalLoopIsotope}}{20}{subsection.4.10.2}
\contentsline {subsection}{\numberline {4.10.3}\leavevmode {\color {Chapter }IntoLoop}}{20}{subsection.4.10.3}
\contentsline {subsection}{\numberline {4.10.4}\leavevmode {\color {Chapter }IntoGroup}}{20}{subsection.4.10.4}
\contentsline {section}{\numberline {4.11}\leavevmode {\color {Chapter }Products of Quasigroups and Loops}}{21}{section.4.11}
\contentsline {subsection}{\numberline {4.11.1}\leavevmode {\color {Chapter }DirectProduct}}{21}{subsection.4.11.1}
\contentsline {section}{\numberline {4.12}\leavevmode {\color {Chapter }Opposite Quasigroups and Loops}}{21}{section.4.12}
\contentsline {subsection}{\numberline {4.12.1}\leavevmode {\color {Chapter }Opposite, OppositeQuasigroup and OppositeLoop}}{21}{subsection.4.12.1}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter }Basic Methods And Attributes}}{22}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter }Basic Attributes}}{22}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }Elements}}{22}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }CayleyTable}}{22}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }One}}{22}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }Size}}{22}{subsection.5.1.4}
\contentsline {subsection}{\numberline {5.1.5}\leavevmode {\color {Chapter }Exponent}}{23}{subsection.5.1.5}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter }Basic Arithmetic Operations}}{23}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }LeftDivision and RightDivision}}{23}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }LeftDivisionCayleyTable and RightDivisionCayleyTable}}{23}{subsection.5.2.2}
\contentsline {section}{\numberline {5.3}\leavevmode {\color {Chapter }Powers and Inverses}}{23}{section.5.3}
\contentsline {subsection}{\numberline {5.3.1}\leavevmode {\color {Chapter }LeftInverse, RightInverse and Inverse}}{24}{subsection.5.3.1}
\contentsline {section}{\numberline {5.4}\leavevmode {\color {Chapter }Associators and Commutators}}{24}{section.5.4}
\contentsline {subsection}{\numberline {5.4.1}\leavevmode {\color {Chapter }Associator}}{24}{subsection.5.4.1}
\contentsline {subsection}{\numberline {5.4.2}\leavevmode {\color {Chapter }Commutator}}{24}{subsection.5.4.2}
\contentsline {section}{\numberline {5.5}\leavevmode {\color {Chapter }Generators}}{24}{section.5.5}
\contentsline {subsection}{\numberline {5.5.1}\leavevmode {\color {Chapter }GeneratorsOfQuasigroup and GeneratorsOfLoop}}{24}{subsection.5.5.1}
\contentsline {subsection}{\numberline {5.5.2}\leavevmode {\color {Chapter }GeneratorsSmallest}}{25}{subsection.5.5.2}
\contentsline {subsection}{\numberline {5.5.3}\leavevmode {\color {Chapter }SmallGeneratingSet}}{25}{subsection.5.5.3}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter }Methods Based on Permutation Groups}}{26}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter }Parent of a Quasigroup}}{26}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }Parent}}{26}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }Position}}{26}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }PosInParent}}{27}{subsection.6.1.3}
\contentsline {section}{\numberline {6.2}\leavevmode {\color {Chapter }Subquasigroups and Subloops}}{27}{section.6.2}
\contentsline {subsection}{\numberline {6.2.1}\leavevmode {\color {Chapter }Subquasigroup}}{27}{subsection.6.2.1}
\contentsline {subsection}{\numberline {6.2.2}\leavevmode {\color {Chapter }Subloop}}{27}{subsection.6.2.2}
\contentsline {subsection}{\numberline {6.2.3}\leavevmode {\color {Chapter }IsSubquasigroup and IsSubloop}}{27}{subsection.6.2.3}
\contentsline {subsection}{\numberline {6.2.4}\leavevmode {\color {Chapter }AllSubquasigroups}}{27}{subsection.6.2.4}
\contentsline {subsection}{\numberline {6.2.5}\leavevmode {\color {Chapter }AllSubloops}}{28}{subsection.6.2.5}
\contentsline {subsection}{\numberline {6.2.6}\leavevmode {\color {Chapter }RightCosets}}{28}{subsection.6.2.6}
\contentsline {subsection}{\numberline {6.2.7}\leavevmode {\color {Chapter }RightTransversal}}{28}{subsection.6.2.7}
\contentsline {section}{\numberline {6.3}\leavevmode {\color {Chapter }Translations and Sections}}{28}{section.6.3}
\contentsline {subsection}{\numberline {6.3.1}\leavevmode {\color {Chapter }LeftTranslation and RightTranslation}}{28}{subsection.6.3.1}
\contentsline {subsection}{\numberline {6.3.2}\leavevmode {\color {Chapter }LeftSection and RightSection}}{28}{subsection.6.3.2}
\contentsline {section}{\numberline {6.4}\leavevmode {\color {Chapter }Multiplication Groups}}{29}{section.6.4}
\contentsline {subsection}{\numberline {6.4.1}\leavevmode {\color {Chapter }LeftMutliplicationGroup, RightMultiplicationGroup and MultiplicationGroup}}{29}{subsection.6.4.1}
\contentsline {subsection}{\numberline {6.4.2}\leavevmode {\color {Chapter }RelativeLeftMultiplicationGroup, RelativeRightMultiplicationGroup and RelativeMultiplicationGroup}}{29}{subsection.6.4.2}
\contentsline {section}{\numberline {6.5}\leavevmode {\color {Chapter }Inner Mapping Groups}}{29}{section.6.5}
\contentsline {subsection}{\numberline {6.5.1}\leavevmode {\color {Chapter }LeftInnerMapping, RightInnerMapping, MiddleInnerMapping}}{30}{subsection.6.5.1}
\contentsline {subsection}{\numberline {6.5.2}\leavevmode {\color {Chapter }LeftInnerMappingGroup, RightInnerMappingGroup, MiddleInnerMappingGroup}}{30}{subsection.6.5.2}
\contentsline {subsection}{\numberline {6.5.3}\leavevmode {\color {Chapter }InnerMappingGroup}}{30}{subsection.6.5.3}
\contentsline {section}{\numberline {6.6}\leavevmode {\color {Chapter }Nuclei, Commutant, Center, and Associator Subloop}}{30}{section.6.6}
\contentsline {subsection}{\numberline {6.6.1}\leavevmode {\color {Chapter }LeftNucles, MiddleNucleus, and RightNucleus}}{30}{subsection.6.6.1}
\contentsline {subsection}{\numberline {6.6.2}\leavevmode {\color {Chapter }Nuc, NucleusOfQuasigroup and NucleusOfLoop}}{31}{subsection.6.6.2}
\contentsline {subsection}{\numberline {6.6.3}\leavevmode {\color {Chapter }Commutant}}{31}{subsection.6.6.3}
\contentsline {subsection}{\numberline {6.6.4}\leavevmode {\color {Chapter }Center}}{31}{subsection.6.6.4}
\contentsline {subsection}{\numberline {6.6.5}\leavevmode {\color {Chapter }AssociatorSubloop}}{31}{subsection.6.6.5}
\contentsline {section}{\numberline {6.7}\leavevmode {\color {Chapter }Normal Subloops and Simple Loops}}{31}{section.6.7}
\contentsline {subsection}{\numberline {6.7.1}\leavevmode {\color {Chapter }IsNormal}}{31}{subsection.6.7.1}
\contentsline {subsection}{\numberline {6.7.2}\leavevmode {\color {Chapter }NormalClosure}}{31}{subsection.6.7.2}
\contentsline {subsection}{\numberline {6.7.3}\leavevmode {\color {Chapter }IsSimple}}{32}{subsection.6.7.3}
\contentsline {section}{\numberline {6.8}\leavevmode {\color {Chapter }Factor Loops}}{32}{section.6.8}
\contentsline {subsection}{\numberline {6.8.1}\leavevmode {\color {Chapter }FactorLoop}}{32}{subsection.6.8.1}
\contentsline {subsection}{\numberline {6.8.2}\leavevmode {\color {Chapter }NaturalHomomorphismByNormalSubloop}}{32}{subsection.6.8.2}
\contentsline {section}{\numberline {6.9}\leavevmode {\color {Chapter }Nilpotency and Central Series}}{32}{section.6.9}
\contentsline {subsection}{\numberline {6.9.1}\leavevmode {\color {Chapter }IsNilpotent}}{32}{subsection.6.9.1}
\contentsline {subsection}{\numberline {6.9.2}\leavevmode {\color {Chapter }NilpotencyClassOfLoop}}{32}{subsection.6.9.2}
\contentsline {subsection}{\numberline {6.9.3}\leavevmode {\color {Chapter }IsStronglyNilpotent}}{32}{subsection.6.9.3}
\contentsline {subsection}{\numberline {6.9.4}\leavevmode {\color {Chapter }UpperCentralSeries}}{33}{subsection.6.9.4}
\contentsline {subsection}{\numberline {6.9.5}\leavevmode {\color {Chapter }LowerCentralSeries}}{33}{subsection.6.9.5}
\contentsline {section}{\numberline {6.10}\leavevmode {\color {Chapter }Solvability, Derived Series and Frattini Subloop}}{33}{section.6.10}
\contentsline {subsection}{\numberline {6.10.1}\leavevmode {\color {Chapter }IsSolvable}}{33}{subsection.6.10.1}
\contentsline {subsection}{\numberline {6.10.2}\leavevmode {\color {Chapter }DerivedSubloop}}{33}{subsection.6.10.2}
\contentsline {subsection}{\numberline {6.10.3}\leavevmode {\color {Chapter }DerivedLength}}{33}{subsection.6.10.3}
\contentsline {subsection}{\numberline {6.10.4}\leavevmode {\color {Chapter }FrattiniSubloop and FrattinifactorSize}}{33}{subsection.6.10.4}
\contentsline {subsection}{\numberline {6.10.5}\leavevmode {\color {Chapter }FrattinifactorSize}}{33}{subsection.6.10.5}
\contentsline {section}{\numberline {6.11}\leavevmode {\color {Chapter }Isomorphisms and Automorphisms}}{33}{section.6.11}
\contentsline {subsection}{\numberline {6.11.1}\leavevmode {\color {Chapter }IsomorphismQuasigroups}}{33}{subsection.6.11.1}
\contentsline {subsection}{\numberline {6.11.2}\leavevmode {\color {Chapter }IsomorphismLoops}}{34}{subsection.6.11.2}
\contentsline {subsection}{\numberline {6.11.3}\leavevmode {\color {Chapter }QuasigroupsUpToIsomorphism}}{34}{subsection.6.11.3}
\contentsline {subsection}{\numberline {6.11.4}\leavevmode {\color {Chapter }LoopsUpToIsomorphism}}{34}{subsection.6.11.4}
\contentsline {subsection}{\numberline {6.11.5}\leavevmode {\color {Chapter }AutomorphismGroup}}{34}{subsection.6.11.5}
\contentsline {subsection}{\numberline {6.11.6}\leavevmode {\color {Chapter }IsomorphicCopyByPerm}}{34}{subsection.6.11.6}
\contentsline {subsection}{\numberline {6.11.7}\leavevmode {\color {Chapter }IsomorphicCopyByNormalSubloop}}{34}{subsection.6.11.7}
\contentsline {subsection}{\numberline {6.11.8}\leavevmode {\color {Chapter }Discriminator}}{35}{subsection.6.11.8}
\contentsline {subsection}{\numberline {6.11.9}\leavevmode {\color {Chapter }AreEqualDiscriminators}}{35}{subsection.6.11.9}
\contentsline {section}{\numberline {6.12}\leavevmode {\color {Chapter }Isotopisms}}{35}{section.6.12}
\contentsline {subsection}{\numberline {6.12.1}\leavevmode {\color {Chapter }IsotopismLoops}}{35}{subsection.6.12.1}
\contentsline {subsection}{\numberline {6.12.2}\leavevmode {\color {Chapter }LoopsUpToIsotopism}}{35}{subsection.6.12.2}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter }Testing Properties of Quasigroups and Loops}}{36}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter }Associativity, Commutativity and Generalizations}}{36}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }IsAssociative}}{36}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }IsCommutative}}{36}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}\leavevmode {\color {Chapter }IsPowerAssociative}}{36}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}\leavevmode {\color {Chapter }IsDiassociative}}{36}{subsection.7.1.4}
\contentsline {section}{\numberline {7.2}\leavevmode {\color {Chapter }Inverse Propeties}}{37}{section.7.2}
\contentsline {subsection}{\numberline {7.2.1}\leavevmode {\color {Chapter }HasLeftInverseProperty, HasRightInverseProperty and HasInverseProperty}}{37}{subsection.7.2.1}
\contentsline {subsection}{\numberline {7.2.2}\leavevmode {\color {Chapter }HasTwosidedInverses}}{37}{subsection.7.2.2}
\contentsline {subsection}{\numberline {7.2.3}\leavevmode {\color {Chapter }HasWeakInverseProperty}}{37}{subsection.7.2.3}
\contentsline {subsection}{\numberline {7.2.4}\leavevmode {\color {Chapter }HasAutomorphicInverseProperty}}{37}{subsection.7.2.4}
\contentsline {subsection}{\numberline {7.2.5}\leavevmode {\color {Chapter }HasAntiautomorphicInverseProperty}}{37}{subsection.7.2.5}
\contentsline {section}{\numberline {7.3}\leavevmode {\color {Chapter }Some Properties of Quasigroups}}{38}{section.7.3}
\contentsline {subsection}{\numberline {7.3.1}\leavevmode {\color {Chapter }IsSemisymmetric}}{38}{subsection.7.3.1}
\contentsline {subsection}{\numberline {7.3.2}\leavevmode {\color {Chapter }IsTotallySymmetric}}{38}{subsection.7.3.2}
\contentsline {subsection}{\numberline {7.3.3}\leavevmode {\color {Chapter }IsIdempotent}}{38}{subsection.7.3.3}
\contentsline {subsection}{\numberline {7.3.4}\leavevmode {\color {Chapter }IsSteinerQuasigroup}}{38}{subsection.7.3.4}
\contentsline {subsection}{\numberline {7.3.5}\leavevmode {\color {Chapter }IsUnipotent}}{38}{subsection.7.3.5}
\contentsline {subsection}{\numberline {7.3.6}\leavevmode {\color {Chapter }IsLeftDistributive, IsRightDistributive, IsDistributive}}{38}{subsection.7.3.6}
\contentsline {subsection}{\numberline {7.3.7}\leavevmode {\color {Chapter }IsEntropic and IsMedial}}{39}{subsection.7.3.7}
\contentsline {section}{\numberline {7.4}\leavevmode {\color {Chapter }Loops of Bol Moufang Type}}{39}{section.7.4}
\contentsline {subsection}{\numberline {7.4.1}\leavevmode {\color {Chapter }IsExtraLoop}}{40}{subsection.7.4.1}
\contentsline {subsection}{\numberline {7.4.2}\leavevmode {\color {Chapter }IsMoufangLoop}}{40}{subsection.7.4.2}
\contentsline {subsection}{\numberline {7.4.3}\leavevmode {\color {Chapter }IsCLoop}}{40}{subsection.7.4.3}
\contentsline {subsection}{\numberline {7.4.4}\leavevmode {\color {Chapter }IsLeftBolLoop}}{40}{subsection.7.4.4}
\contentsline {subsection}{\numberline {7.4.5}\leavevmode {\color {Chapter }IsRightBolLoop}}{40}{subsection.7.4.5}
\contentsline {subsection}{\numberline {7.4.6}\leavevmode {\color {Chapter }IsLCLoop}}{40}{subsection.7.4.6}
\contentsline {subsection}{\numberline {7.4.7}\leavevmode {\color {Chapter }IsRCLoop}}{40}{subsection.7.4.7}
\contentsline {subsection}{\numberline {7.4.8}\leavevmode {\color {Chapter }IsLeftNuclearSquareLoop}}{40}{subsection.7.4.8}
\contentsline {subsection}{\numberline {7.4.9}\leavevmode {\color {Chapter }IsMiddleNuclearSquareLoop}}{40}{subsection.7.4.9}
\contentsline {subsection}{\numberline {7.4.10}\leavevmode {\color {Chapter }IsRightNuclearSquareLoop}}{40}{subsection.7.4.10}
\contentsline {subsection}{\numberline {7.4.11}\leavevmode {\color {Chapter }IsNuclearSquareLoop}}{41}{subsection.7.4.11}
\contentsline {subsection}{\numberline {7.4.12}\leavevmode {\color {Chapter }IsFlexible}}{41}{subsection.7.4.12}
\contentsline {subsection}{\numberline {7.4.13}\leavevmode {\color {Chapter }IsLeftAlternative}}{41}{subsection.7.4.13}
\contentsline {subsection}{\numberline {7.4.14}\leavevmode {\color {Chapter }IsRightAlternative}}{41}{subsection.7.4.14}
\contentsline {subsection}{\numberline {7.4.15}\leavevmode {\color {Chapter }IsAlternative}}{41}{subsection.7.4.15}
\contentsline {section}{\numberline {7.5}\leavevmode {\color {Chapter }Power Alternative Loops}}{42}{section.7.5}
\contentsline {subsection}{\numberline {7.5.1}\leavevmode {\color {Chapter }IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative}}{42}{subsection.7.5.1}
\contentsline {section}{\numberline {7.6}\leavevmode {\color {Chapter }Conjugacy Closed Loops and Related Properties}}{42}{section.7.6}
\contentsline {subsection}{\numberline {7.6.1}\leavevmode {\color {Chapter }IsLCCLoop}}{42}{subsection.7.6.1}
\contentsline {subsection}{\numberline {7.6.2}\leavevmode {\color {Chapter }IsRCCLoop}}{42}{subsection.7.6.2}
\contentsline {subsection}{\numberline {7.6.3}\leavevmode {\color {Chapter }IsCCLoop}}{42}{subsection.7.6.3}
\contentsline {subsection}{\numberline {7.6.4}\leavevmode {\color {Chapter }IsOsbornLoop}}{42}{subsection.7.6.4}
\contentsline {section}{\numberline {7.7}\leavevmode {\color {Chapter }Automorphic Loops}}{43}{section.7.7}
\contentsline {subsection}{\numberline {7.7.1}\leavevmode {\color {Chapter }IsLeftAutomorphicLoop}}{43}{subsection.7.7.1}
\contentsline {subsection}{\numberline {7.7.2}\leavevmode {\color {Chapter }IsMiddleAutomorphicLoop}}{43}{subsection.7.7.2}
\contentsline {subsection}{\numberline {7.7.3}\leavevmode {\color {Chapter }IsRightAutomorphicLoop}}{44}{subsection.7.7.3}
\contentsline {subsection}{\numberline {7.7.4}\leavevmode {\color {Chapter }IsAutomorphicLoop}}{44}{subsection.7.7.4}
\contentsline {section}{\numberline {7.8}\leavevmode {\color {Chapter }Additonal Varieties of Loops}}{44}{section.7.8}
\contentsline {subsection}{\numberline {7.8.1}\leavevmode {\color {Chapter }IsCodeLoop}}{44}{subsection.7.8.1}
\contentsline {subsection}{\numberline {7.8.2}\leavevmode {\color {Chapter }IsSteinerLoop}}{44}{subsection.7.8.2}
\contentsline {subsection}{\numberline {7.8.3}\leavevmode {\color {Chapter }IsLeftBruckLoop and IsLeftKLoop}}{44}{subsection.7.8.3}
\contentsline {subsection}{\numberline {7.8.4}\leavevmode {\color {Chapter }IsRightBruckLoop and IsRightKLoop}}{44}{subsection.7.8.4}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter }Specific Methods}}{45}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter }Core Methods for Bol Loops}}{45}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }AssociatedLeftBruckLoop and AssociatedRightBruckLoop}}{45}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }IsExactGroupFactorization}}{45}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }RightBolLoopByExactGroupFactorization}}{45}{subsection.8.1.3}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter }Moufang Modifications}}{46}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }LoopByCyclicModification}}{46}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }LoopByDihedralModification}}{46}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }LoopMG2}}{46}{subsection.8.2.3}
\contentsline {section}{\numberline {8.3}\leavevmode {\color {Chapter }Triality for Moufang Loops}}{46}{section.8.3}
\contentsline {subsection}{\numberline {8.3.1}\leavevmode {\color {Chapter }TrialityPermGroup}}{47}{subsection.8.3.1}
\contentsline {subsection}{\numberline {8.3.2}\leavevmode {\color {Chapter }TrialityPcGroup}}{47}{subsection.8.3.2}
\contentsline {section}{\numberline {8.4}\leavevmode {\color {Chapter }Realizing Groups as Multiplication Groups of Loops}}{47}{section.8.4}
\contentsline {subsection}{\numberline {8.4.1}\leavevmode {\color {Chapter }AllLoopTablesInGroup}}{47}{subsection.8.4.1}
\contentsline {subsection}{\numberline {8.4.2}\leavevmode {\color {Chapter }AllProperLoopTablesInGroup}}{47}{subsection.8.4.2}
\contentsline {subsection}{\numberline {8.4.3}\leavevmode {\color {Chapter }OneLoopTableInGroup}}{47}{subsection.8.4.3}
\contentsline {subsection}{\numberline {8.4.4}\leavevmode {\color {Chapter }OneProperLoopTableInGroup}}{48}{subsection.8.4.4}
\contentsline {subsection}{\numberline {8.4.5}\leavevmode {\color {Chapter }AllLoopsWithMltGroup}}{48}{subsection.8.4.5}
\contentsline {subsection}{\numberline {8.4.6}\leavevmode {\color {Chapter }OneLoopWithMltGroup}}{48}{subsection.8.4.6}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter }Libraries of Loops}}{49}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter }A Typical Library}}{49}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }LibraryLoop}}{49}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}\leavevmode {\color {Chapter }MyLibraryLoop}}{49}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}\leavevmode {\color {Chapter }DisplayLibraryInfo}}{50}{subsection.9.1.3}
\contentsline {section}{\numberline {9.2}\leavevmode {\color {Chapter }Left Bol Loops and Right Bol Loops}}{50}{section.9.2}
\contentsline {subsection}{\numberline {9.2.1}\leavevmode {\color {Chapter }LeftBolLoop}}{50}{subsection.9.2.1}
\contentsline {subsection}{\numberline {9.2.2}\leavevmode {\color {Chapter }RightBolLoop}}{50}{subsection.9.2.2}
\contentsline {section}{\numberline {9.3}\leavevmode {\color {Chapter }Moufang Loops}}{50}{section.9.3}
\contentsline {subsection}{\numberline {9.3.1}\leavevmode {\color {Chapter }MoufangLoop}}{50}{subsection.9.3.1}
\contentsline {section}{\numberline {9.4}\leavevmode {\color {Chapter }Code Loops}}{51}{section.9.4}
\contentsline {subsection}{\numberline {9.4.1}\leavevmode {\color {Chapter }CodeLoop}}{51}{subsection.9.4.1}
\contentsline {section}{\numberline {9.5}\leavevmode {\color {Chapter }Steiner Loops}}{51}{section.9.5}
\contentsline {subsection}{\numberline {9.5.1}\leavevmode {\color {Chapter }SteinerLoop}}{51}{subsection.9.5.1}
\contentsline {section}{\numberline {9.6}\leavevmode {\color {Chapter }Conjugacy Closed Loops}}{51}{section.9.6}
\contentsline {subsection}{\numberline {9.6.1}\leavevmode {\color {Chapter }RCCLoop and RightConjugacyClosedLoop}}{52}{subsection.9.6.1}
\contentsline {subsection}{\numberline {9.6.2}\leavevmode {\color {Chapter }LCCLoop and LeftConjugacyClosedLoop}}{52}{subsection.9.6.2}
\contentsline {subsection}{\numberline {9.6.3}\leavevmode {\color {Chapter }CCLoop and ConjugacyClosedLoop}}{52}{subsection.9.6.3}
\contentsline {section}{\numberline {9.7}\leavevmode {\color {Chapter }Small Loops}}{52}{section.9.7}
\contentsline {subsection}{\numberline {9.7.1}\leavevmode {\color {Chapter }SmallLoop}}{53}{subsection.9.7.1}
\contentsline {section}{\numberline {9.8}\leavevmode {\color {Chapter }Paige Loops}}{53}{section.9.8}
\contentsline {subsection}{\numberline {9.8.1}\leavevmode {\color {Chapter }PaigeLoop}}{53}{subsection.9.8.1}
\contentsline {section}{\numberline {9.9}\leavevmode {\color {Chapter }Nilpotent Loops}}{53}{section.9.9}
\contentsline {subsection}{\numberline {9.9.1}\leavevmode {\color {Chapter }NilpotentLoop}}{53}{subsection.9.9.1}
\contentsline {section}{\numberline {9.10}\leavevmode {\color {Chapter }Automorphic Loops}}{53}{section.9.10}
\contentsline {subsection}{\numberline {9.10.1}\leavevmode {\color {Chapter }AutomorphicLoop}}{53}{subsection.9.10.1}
\contentsline {section}{\numberline {9.11}\leavevmode {\color {Chapter }Interesting Loops}}{54}{section.9.11}
\contentsline {subsection}{\numberline {9.11.1}\leavevmode {\color {Chapter }InterestingLoop}}{54}{subsection.9.11.1}
\contentsline {section}{\numberline {9.12}\leavevmode {\color {Chapter }Libraries of Loops Up To Isotopism}}{54}{section.9.12}
\contentsline {subsection}{\numberline {9.12.1}\leavevmode {\color {Chapter }ItpSmallLoop}}{54}{subsection.9.12.1}
\contentsline {chapter}{\numberline {A}\leavevmode {\color {Chapter }Files}}{55}{appendix.A}
\contentsline {chapter}{\numberline {B}\leavevmode {\color {Chapter }Filters}}{57}{appendix.B}
\contentsline {chapter}{References}{61}{appendix*.3}
\contentsline {chapter}{Index}{62}{section*.4}

64
doc/loops.xml
View File

@ -6,7 +6,8 @@
<!-- Read the file pkg/loops/etc/gapdoc.txt for instructions on how to produce the documentation. -->
<!-- Typesetting rules for this document that produce acceptable outcome in both html and pdf:
<!-- Typesetting rules for this document that produce acceptable outcome in
both html and pdf:
* Use <Br/><P/> at the beginning of paragraphs that are supposed to have an empty line just before them for greater emphasis, eg., before Remark, Example, and to break up the text. (This has no effect on html.)
@ -24,7 +25,7 @@
<TitlePage>
<Title>The <Package>LOOPS</Package> Package</Title>
<Version>Version 3.3.0</Version>
<Version>Version 3.4.0</Version>
<Subtitle>Computing with quasigroups and loops in &GAP;</Subtitle>
<Author>G&#225;bor P. Nagy
<Email>nagyg@math.u-szeged.hu</Email>
@ -34,7 +35,7 @@
<Email>petr@math.du.edu</Email>
<Address>Department of Mathematics, University of Denver</Address>
</Author>
<Copyright>&copyright; 2016 G&#225;bor P. Nagy and Petr Vojt&#283;chovsk&#253;.
<Copyright>&copyright; 2017 G&#225;bor P. Nagy and Petr Vojt&#283;chovsk&#253;.
</Copyright>
</TitlePage>
@ -65,7 +66,7 @@
<Section Label="Sec:Installation"> <Heading>Installation</Heading>
Have <Package>GAP 4.7</Package> or newer installed on your computer.
Have <Package>GAP 4.8</Package> or newer installed on your computer.
<P/>If you do not see the subfolder <File>pkg/loops</File> in the main directory of &GAP; then download the <Package>LOOPS</Package> package from the distribution website <URL>http://www.math.du.edu/loops</URL> and unpack the downloaded file into the <File>pkg</File> subfolder.
@ -124,9 +125,9 @@ We welcome all comments and suggestions on <Package>LOOPS</Package>, especially
<Section Label="Sec:Acknowledgment"> <Heading>Acknowledgment</Heading>
We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Ale&#353; Dr&#225;pal, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank L&#252;beck and Jonathan D.H. Smith.
We thank the following people for sending us remarks and comments, and for suggesting new functionality of the package: Muniru Asiru, Bjoern Assmann, Andreas Distler, Ale&#353; Dr&#225;pal, Graham Ellis, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank L&#252;beck, Jonathan D.H. Smith, David Stanovsk&#253; and Glen Whitney.
<P/>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of commutative automorphic loops of order 27, 81 and 243 was obtained jointly with Izabella Stuhl.
<P/>The library of Moufang loops of order 243 was generated from data provided by Michael C. Slattery and Ashley L. Zenisek. The library of right conjugacy closed loops of order less than 28 was generated from data provided by Katharina Artic. The library of right Bruck loops of order 27, 81 was obtained jointly with Izabella Stuhl.
<P/>G&#225;bor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr Vojt&#283;chovsk&#253; was supported by the 2006 and 2016 University of Denver PROF grants and the Simons Foundation Collaboration Grant 210176.
@ -260,8 +261,8 @@ DeclareCategory( "IsLoopElement",
DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsLatinMagma", IsObject );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
DeclareCategory( "IsLoop", IsQuasigroup and
IsMultiplicativeElementWithInverseCollection);
</Verb>
@ -1325,9 +1326,19 @@ See Section <Ref Sect="Sec:NilpotenceAndSolvability"/> for definitions of solvab
<Br/><P/>While dealing with Cayley tables, it is often useful to rename or reorder the elements of the underlying quasigroup without changing the isomorphism type of the quasigroups. <Package>LOOPS</Package> contains several functions for this purpose.
<ManSection>
<Oper Name="QuasigroupIsomorph" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a quasigroup and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M>, returns the quasigroup defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>.</Returns>
</ManSection>
<ManSection>
<Oper Name="LoopIsomorph" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a loop and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M> fixing <M>1</M>, returns the loop defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>. If <Arg>f</Arg><M>(1)=c\ne 1</M>, the isomorphism <M>(1,c)</M> is applied after <Arg>f</Arg>.</Returns>
</ManSection>
<ManSection>
<Oper Name="IsomorphicCopyByPerm" Arg="Q, f"/>
<Returns>When <Arg>Q</Arg> is a quasigroup and <Arg>f</Arg> is a permutation of <M>1,\dots,|</M><Arg>Q</Arg><M>|</M>, returns a quasigroup defined on the same set as <Arg>Q</Arg> with multiplication <M>*</M> defined by <M>x*y = </M><Arg>f</Arg><M>(</M><Arg>f</Arg><M>{}^{-1}(x)</M><Arg>f</Arg><M>{}^{-1}(y))</M>. When <Arg>Q</Arg> is a declared loop, a loop is returned. Consequently, when <Arg>Q</Arg> is a declared loop and <Arg>f</Arg><M>(1) = k\ne 1</M>, then <Arg>f</Arg> is first replaced with <Arg>f</Arg><M>\circ (1,k)</M>, to make sure that the resulting Cayley table is normalized.</Returns>
<Returns><Code>LoopIsomorphism(<Arg>Q</Arg>,<Arg>f</Arg>)</Code> if <Arg>Q</Arg> is a loop, and <Code>QuasigroupIsomorphism(<Arg>Q</Arg>,<Arg>f</Arg>)</Code> if <Arg>Q</Arg> is a quasigroup.</Returns>
</ManSection>
<ManSection>
@ -1971,6 +1982,26 @@ The library named <Emph>left Bol</Emph> contains all nonassociative left Bol loo
</Section>
<!-- Section: Left Bruck Loops and Right Bruck Loops --------------------------------------------------- -->
<Section Label="Sec:BruckLoops"> <Heading>Left Bruck Loops and Right Bruck Loops</Heading>
The emmerging library named <Emph>left Bruck</Emph> contains all left Bruck loops of orders <M>3</M>, <M>9</M>, <M>27</M> and <M>81</M> (there are <M>1</M>, <M>2</M>, <M>7</M> and <M>72</M> such loops, respectively).
<P/>For an odd prime <M>p</M>, left Bruck loops of order <M>p^k</M> are centrally nilpotent and hence central extensions of the cyclic group of order <M>p</M> by a left Bruck loop of order <M>p^{k-1}</M>. It is known that left Bruck loops of order <M>p</M> and <M>p^2</M> are abelian groups; we have included them in the library because of the iterative nature of the construction of nilpotent loops.
<ManSection>
<Func Name="LeftBruckLoop" Arg="n, m"/>
<Returns>The <Arg>m</Arg>th left Bruck loop of order <Arg>n</Arg> in the library.</Returns>
</ManSection>
<ManSection>
<Func Name="RightBruckLoop" Arg="n, m"/>
<Returns>The <Arg>m</Arg>th right Bruck loop of order <Arg>n</Arg> in the library.</Returns>
</ManSection>
</Section>
<!-- Section: Moufang Loops ---------------------------------------------------------------------------- -->
<Section Label="Sec:MoufangLoops"> <Heading>Moufang Loops</Heading>
@ -2072,7 +2103,7 @@ The following table summarizes the number of right conjugacy closed loops of a g
<Description><B>Remark:</B> Only the right conjugacy closed loops are stored in the library. Left conjugacy closed loops are obtained from right conjugacy closed loops via <Code>Opposite</Code>.<Br/></Description>
</ManSection>
<P/>The library named <Emph>CC</Emph> contains all nonassociative conjugacy closed loops of order <M>n\le 27</M> and also of orders <M>2p</M> and <M>p^2</M> for all primes <M>p</M>.
<P/>The library named <Emph>CC</Emph> contains all CC loops of order <M>2\le 2^k\le 64</M>, <M>3\le 3^k\le 81</M>, <M>5\le 5^k\le 125</M>, <M>7\le 7^k\le 343</M>, all nonassociative CC loops of order less than 28, and all nonassociative CC loops of order <M>p^2</M> and <M>2p</M> for any odd prime <M>p</M>.
<P/>By results of Kunen <Cite Key="Kun"/>, for every odd prime <M>p</M> there are precisely 3 nonassociative conjugacy closed loops of order <M>p^2</M>. Cs&#246;rg&#337; and Dr&#225;pal <Cite Key="CsDr"/> described these 3 loops by multiplicative formulas on <M>\mathbb{Z}_{p^2}</M> and <M>\mathbb{Z}_p \times \mathbb{Z}_p</M> as follows:
<List>
@ -2141,7 +2172,9 @@ are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 nilpote
<Section Label="Sec:AutomorphicLoops"> <Heading>Automorphic Loops</Heading>
The library named <Emph>automorphic</Emph> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15), all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops, respectively, including abelian groups), and commutative automorphic loops <M>Q</M> of order 243 possessing a central subloop <M>S</M> of order 3 such that <M>Q/S</M> is not the elementary abelian group of order 81 (there are 118451 such loops).
The library named <Emph>automorphic</Emph> contains all nonassociative automorphic loops of order less than 16 up to isomorphism (there is 1 such loop of order 6, 7 of order 8, 3 of order 10, 2 of order 12, 5 of order 14, and 2 of order 15) and all commutative automorphic loops of order 3, 9, 27 and 81 (there are 1, 2, 7 and 72 such loops).
<P/>It turns out that commutative automorphic loops of order 3, 9, 27 and 81 (but not 243) are in one-to-on correspondence with left Bruck loops of the respective orders, see <Cite Key="Greer"/>, <Cite Key="StuhlVojtechovsky"/>. Only the left Bruck loops are stored in the library.
<ManSection>
<Func Name="AutomorphicLoop" Arg="n, m"/>
@ -2343,9 +2376,6 @@ Many implications among properties of loops are built directly into <Package>LOO
<Br/><Code>( IsLeftAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsAutomorphicLoop )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsCommutative )</Code>
<Br/><Code>( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )</Code>
<Br/><Code>( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and HasAntiautomorphicInverseProperty )</Code>
<Br/><Code>( IsFlexible, IsMiddleAutomorphicLoop )</Code>
@ -2356,11 +2386,15 @@ Many implications among properties of loops are built directly into <Package>LOO
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )</Code>
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )</Code>
<Br/><Code>( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )</Code>
<Br/><Code>( IsMiddleAutomorphicLoop, IsCommutative )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLeftBruckLoop )</Code>
<Br/><Code>( IsLeftAutomorphicLoop, IsLCCLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRightBruckLoop )</Code>
<Br/><Code>( IsRightAutomorphicLoop, IsRCCLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )</Code>
<Br/><Code>( IsAutomorphicLoop, IsAssociative )</Code>
</Appendix>
@ -2368,4 +2402,4 @@ Many implications among properties of loops are built directly into <Package>LOO
<TheIndex/>
</Book>
</Book>

559
doc/loops_bib.xml
View File

@ -1,559 +0,0 @@
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE file SYSTEM "bibxmlext.dtd">
<file>
<entry id="Artic"><phdthesis>
<author>
<name><first>Katharina</first><last>Artic</last></name>
</author>
<title>On conjugacy closed loops and conjugacy closed loop folders</title>
<school>RWTH Aachen University</school>
<year>2015</year>
</phdthesis></entry>
<entry id="Ar"><article>
<author>
<name><first>R.</first><last>Artzy</last></name>
</author>
<title>On automorphic-inverse properties in loops</title>
<journal>Proc. Amer. Math. Soc.</journal>
<year>1959</year>
<volume>10</volume>
<pages>588–591</pages>
<issn>0002-9939</issn>
<mrnumber>0107674 (21 #6397)</mrnumber>
<mrclass>20.00</mrclass>
<mrreviewer>H. Minc</mrreviewer>
<other type="fjournal">Proceedings of the American Mathematical
Society</other>
</article></entry>
<entry id="BaGrVo"><article>
<author>
<name><first>Dylene Agda Souza</first><last>De Barros</last></name>
<name><first>Alexander</first><last>Grishkov</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Commutative automorphic loops of order <C><M>p^3</M></C></title>
<journal>J. Algebra Appl.</journal>
<year>2012</year>
<volume>11</volume>
<number>5</number>
<pages>1250100, 15</pages>
<issn>0219-4988</issn>
<mrnumber>2983192</mrnumber>
<mrclass>20N05 (20G40)</mrclass>
<mrreviewer>Ágota Figula</mrreviewer>
<url>http://dx.doi.org/10.1142/S0219498812501009</url>
<other type="doi">10.1142/S0219498812501009</other>
<other type="fjournal">Journal of Algebra and its Applications</other>
</article></entry>
<entry id="Br"><book>
<author>
<name><first>Richard Hubert</first><last>Bruck</last></name>
</author>
<title>A survey of binary systems</title>
<publisher>Springer Verlag</publisher>
<year>1958</year>
<series>Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge,
Heft 20. Reihe: Gruppentheorie</series>
<address>Berlin</address>
<mrnumber>0093552 (20 #76)</mrnumber>
<mrclass>20.00</mrclass>
<mrreviewer>L. J. Paige</mrreviewer>
<other type="pages">viii+185</other>
</book></entry>
<entry id="BrPa"><article>
<author>
<name><first>R. H.</first><last>Bruck</last></name>
<name><first>Lowell J.</first><last>Paige</last></name>
</author>
<title>Loops whose inner mappings are automorphisms</title>
<journal>Ann. of Math. (2)</journal>
<year>1956</year>
<volume>63</volume>
<pages>308–323</pages>
<issn>0003-486X</issn>
<mrnumber>0076779</mrnumber>
<mrclass>20.0X</mrclass>
<mrreviewer>R. Moufang</mrreviewer>
<other type="fjournal">Annals of Mathematics. Second Series</other>
</article></entry>
<entry id="ChPfSm"><book>
<editor>
<name><first>O.</first><last>Chein</last></name>
<name><first>H. O.</first><last>Pflugfelder</last></name>
<name><first>J. D. H.</first><last>Smith</last></name>
</editor>
<title>Quasigroups and loops: theory and applications</title>
<publisher>Heldermann Verlag</publisher>
<year>1990</year>
<volume>8</volume>
<series>Sigma Series in Pure Mathematics</series>
<address>Berlin</address>
<isbn>3-88538-008-0</isbn>
<mrnumber>1125806 (93g:20133)</mrnumber>
<mrclass>20N05 (20-06)</mrclass>
<mrreviewer>D. A. Robinson</mrreviewer>
<other type="pages">xii+568</other>
</book></entry>
<entry id="CoRo"><book>
<author>
<name><first>Charles J.</first><last>Colbourn</last></name>
<name><first>Alexander</first><last>Rosa</last></name>
</author>
<title>Triple systems</title>
<publisher>The Clarendon Press Oxford University Press</publisher>
<year>1999</year>
<series>Oxford Mathematical Monographs</series>
<address>New York</address>
<isbn>0-19-853576-7</isbn>
<mrnumber>1843379 (2002h:05024)</mrnumber>
<mrclass>05B07 (05-02)</mrclass>
<mrreviewer>Elizabeth J. Billington</mrreviewer>
<other type="pages">xvi+560</other>
</book></entry>
<entry id="CsDr"><article>
<author>
<name><first>Piroska</first><last>Csörgő</last></name>
<name><first>Aleš</first><last>Drápal</last></name>
</author>
<title>Left conjugacy closed loops of nilpotency class two</title>
<journal>Results Math.</journal>
<year>2005</year>
<volume>47</volume>
<number>3-4</number>
<pages>242–265</pages>
<issn>1422-6383</issn>
<mrnumber>2153496 (2006b:20095)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Huberta Lausch</mrreviewer>
<other type="fjournal">Results in Mathematics. Resultate der
Mathematik</other>
</article></entry>
<entry id="DaVo"><article>
<author>
<name><first>Daniel</first><last>Daly</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Enumeration of nilpotent loops via cohomology</title>
<journal>J. Algebra</journal>
<year>2009</year>
<volume>322</volume>
<number>11</number>
<pages>4080–4098</pages>
<issn>0021-8693</issn>
<mrnumber>2556139 (2011e:20098)</mrnumber>
<mrclass>20N05 (20J99)</mrclass>
<mrreviewer>Yu. M. Movsisyan</mrreviewer>
<url>http://dx.doi.org/10.1016/j.jalgebra.2009.03.042</url>
<other type="coden">JALGA4</other>
<other type="doi">10.1016/j.jalgebra.2009.03.042</other>
<other type="fjournal">Journal of Algebra</other>
</article></entry>
<entry id="DrapalCD"><article>
<author>
<name><first>Aleš</first><last>Drápal</last></name>
</author>
<title>Cyclic and dihedral constructions of even order</title>
<journal>Comment. Math. Univ. Carolin.</journal>
<year>2003</year>
<volume>44</volume>
<number>4</number>
<pages>593–614</pages>
<issn></issn>
<mrnumber>MR2062876 (2005d:20038)</mrnumber>
<mrclass>20D60 (05B15)</mrclass>
<mrreviewer>Thomas Michael Keller</mrreviewer>
<url></url>
<other type="coden"></other>
<other type="doi"></other>
<other type="fjournal">Commentationes Mathematicae Universitatis
Carolinae</other>
</article></entry>
<entry id="DrVo"><article>
<author>
<name><first>Aleš</first><last>Drápal</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Moufang loops that share associator and three quarters of
their multiplication tables</title>
<journal>Rocky Mountain J. Math.</journal>
<year>2006</year>
<volume>36</volume>
<number>2</number>
<pages>425–455</pages>
<issn>0035-7596</issn>
<mrnumber>2234814 (2007d:20114)</mrnumber>
<mrclass>20N05 (05B15)</mrclass>
<mrreviewer>Orin Chein</mrreviewer>
<url>http://dx.doi.org/10.1216/rmjm/1181069461</url>
<other type="coden">RMJMAE</other>
<other type="doi">10.1216/rmjm/1181069461</other>
<other type="fjournal">The Rocky Mountain Journal of Mathematics</other>
</article></entry>
<entry id="Fe"><article>
<author>
<name><first>Ferenc</first><last>Fenyves</last></name>
</author>
<title>Extra loops. <C>II</C>. <C>O</C>n loops with identities of
<C>B</C>ol-<C>M</C>oufang type</title>
<journal>Publ. Math. Debrecen</journal>
<year>1969</year>
<volume>16</volume>
<pages>187–192</pages>
<issn>0033-3883</issn>
<mrnumber>0262409 (41 #7017)</mrnumber>
<mrclass>20.95</mrclass>
<mrreviewer>D. A. Robinson</mrreviewer>
<other type="fjournal">Publicationes Mathematicae Debrecen</other>
</article></entry>
<entry id="Go"><book>
<author>
<name><first>Edgar G.</first><last>Goodaire</last></name>
<name><first>Sean</first><last>May</last></name>
<name><first>Maitreyi</first><last>Raman</last></name>
</author>
<title>The <C>M</C>oufang loops of order less than 64</title>
<publisher>Nova Science Publishers Inc.</publisher>
<year>1999</year>
<address>Commack, NY</address>
<isbn>1-56072-659-8</isbn>
<mrnumber>1689624 (2000a:20147)</mrnumber>
<mrclass>20N05</mrclass>
<other type="pages">xviii+287</other>
</book></entry>
<entry id="GrKiNa"><article>
<author>
<name><first>Alexander</first><last>Grishkov</last></name>
<name><first>Michael</first><last>Kinyon</last></name>
<name><first>Gábor P.</first><last>Nagy</last></name>
</author>
<title>Solvability of commutative automorphic loops</title>
<journal>Proc. Amer. Math. Soc.</journal>
<year>2014</year>
<volume>142</volume>
<number>9</number>
<pages>3029–3037</pages>
<issn>0002-9939</issn>
<mrnumber>3223359</mrnumber>
<mrclass>20N05 (17B99)</mrclass>
<mrreviewer>J. D. Phillips</mrreviewer>
<url>http://dx.doi.org/10.1090/S0002-9939-2014-12053-3</url>
<other type="doi">10.1090/S0002-9939-2014-12053-3</other>
<other type="fjournal">Proceedings of the American Mathematical
Society</other>
</article></entry>
<entry id="JaMa"><article>
<author>
<name><first>Mark T.</first><last>Jacobson</last></name>
<name><first>Peter</first><last>Matthews</last></name>
</author>
<title>Generating uniformly distributed random <C>L</C>atin squares</title>
<journal>J. Combin. Des.</journal>
<year>1996</year>
<volume>4</volume>
<number>6</number>
<pages>405–437</pages>
<issn>1063-8539</issn>
<mrnumber>1410617 (98b:05021)</mrnumber>
<mrclass>05B15 (60J10)</mrclass>
<mrreviewer>Lars Døvling Andersen</mrreviewer>
<url>http://dx.doi.org/10.1002/(SICI)1520-6610(1996)4:6&lt;405::AID-JCD3>3.0.CO;2-J</url>
<other
type="doi">10.1002/(SICI)1520-6610(1996)4:6&lt;405::AID-JCD3>3.0.CO;2-J</other>
<other type="fjournal">Journal of Combinatorial Designs</other>
</article></entry>
<entry id="JeKiVo"><article>
<author>
<name><first>P{\v{r}}emysl</first><last>Jedlička</last></name>
<name><first>Michael</first><last>Kinyon</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Nilpotency in automorphic loops of prime power order</title>
<journal>J. Algebra</journal>
<year>2012</year>
<volume>350</volume>
<pages>64–76</pages>
<issn>0021-8693</issn>
<mrnumber>2859875</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Mohammad Shahryari</mrreviewer>
<url>http://dx.doi.org/10.1016/j.jalgebra.2011.09.034</url>
<other type="coden">JALGA4</other>
<other type="doi">10.1016/j.jalgebra.2011.09.034</other>
<other type="fjournal">Journal of Algebra</other>
</article></entry>
<entry id="JoKiNaVo"><article>
<author>
<name><first>Kenneth W.</first><last>Johnson</last></name>
<name><first>Michael K.</first><last>Kinyon</last></name>
<name><first>Gábor P.</first><last>Nagy</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Searching for small simple automorphic loops</title>
<journal>LMS J. Comput. Math.</journal>
<year>2011</year>
<volume>14</volume>
<pages>200–213</pages>
<issn>1461-1570</issn>
<mrnumber>2831230</mrnumber>
<mrclass>20N05 (20B15 20B40)</mrclass>
<mrreviewer>Tuval S. Foguel</mrreviewer>
<url>http://dx.doi.org/10.1112/S1461157010000173</url>
<other type="doi">10.1112/S1461157010000173</other>
<other type="fjournal">LMS Journal of Computation and Mathematics</other>
</article></entry>
<entry id="KiKuPh"><article>
<author>
<name><first>Michael K.</first><last>Kinyon</last></name>
<name><first>Kenneth</first><last>Kunen</last></name>
<name><first>J. D.</first><last>Phillips</last></name>
</author>
<title>Every diassociative <C><M>A</M></C>-loop is <C>M</C>oufang</title>
<journal>Proc. Amer. Math. Soc.</journal>
<year>2002</year>
<volume>130</volume>
<number>3</number>
<pages>619–624</pages>
<issn>0002-9939</issn>
<mrnumber>1866009 (2002k:20124)</mrnumber>
<mrclass>20N05 (68T15)</mrclass>
<mrreviewer>Orin Chein</mrreviewer>
<url>http://dx.doi.org/10.1090/S0002-9939-01-06090-7</url>
<other type="coden">PAMYAR</other>
<other type="doi">10.1090/S0002-9939-01-06090-7</other>
<other type="fjournal">Proceedings of the American Mathematical
Society</other>
</article></entry>
<entry id="KiKuPhVo"><article>
<author>
<name><first>Michael K.</first><last>Kinyon</last></name>
<name><first>Kenneth</first><last>Kunen</last></name>
<name><first>J. D.</first><last>Phillips</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>The structure of automorphic loops</title>
<journal>Trans. Amer. Math. Soc.</journal>
<year>2016</year>
<volume>368</volume>
<number>12</number>
<pages>8901–8927</pages>
<issn>0002-9947</issn>
<mrnumber>3551593</mrnumber>
<mrclass>20N05</mrclass>
<url>http://dx.doi.org/10.1090/tran/6622</url>
<other type="coden">TAMTAM</other>
<other type="doi">10.1090/tran/6622</other>
<other type="fjournal">Transactions of the American Mathematical
Society</other>
</article></entry>
<entry id="KiNaVo2015"><article>
<author>
<name><first>Michael K.</first><last>Kinyon</last></name>
<name><first>Gábor P.</first><last>Nagy</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Bol loops and Bruck loops of order <M>pq</M></title>
<journal></journal>
<year>2015</year>
<note>preprint</note>
</article></entry>
<entry id="Kun"><article>
<author>
<name><first>Kenneth</first><last>Kunen</last></name>
</author>
<title>The structure of conjugacy closed loops</title>
<journal>Trans. Amer. Math. Soc.</journal>
<year>2000</year>
<volume>352</volume>
<number>6</number>
<pages>2889–2911</pages>
<issn>0002-9947</issn>
<mrnumber>1615991 (2000j:20132)</mrnumber>
<mrclass>20N05 (03C05)</mrclass>
<mrreviewer>Edgar G. Goodaire</mrreviewer>
<url>http://dx.doi.org/10.1090/S0002-9947-00-02350-3</url>
<other type="coden">TAMTAM</other>
<other type="doi">10.1090/S0002-9947-00-02350-3</other>
<other type="fjournal">Transactions of the American Mathematical
Society</other>
</article></entry>
<entry id="Li"><article>
<author>
<name><first>Martin W.</first><last>Liebeck</last></name>
</author>
<title>The classification of finite simple <C>M</C>oufang loops</title>
<journal>Math. Proc. Cambridge Philos. Soc.</journal>
<year>1987</year>
<volume>102</volume>
<number>1</number>
<pages>33–47</pages>
<issn>0305-0041</issn>
<mrnumber>886433 (88g:20146)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Karl H. Robinson</mrreviewer>
<url>http://dx.doi.org/10.1017/S0305004100067025</url>
<other type="coden">MPCPCO</other>
<other type="doi">10.1017/S0305004100067025</other>
<other type="fjournal">Mathematical Proceedings of the Cambridge
Philosophical
Society</other>
</article></entry>
<entry id="Mo"><unpublished>
<author>
<name><first>G. Eric</first><last>Moorhouse</last></name>
</author>
<title>Bol loops of small order</title>
<note>http://www.uwyo.edu/moorhouse/pub/bol/</note>
</unpublished></entry>
<entry id="Na"><article>
<author>
<name><first>Gábor P.</first><last>Nagy</last></name>
</author>
<title>A class of simple proper <C>B</C>ol loops</title>
<journal>Manuscripta Math.</journal>
<year>2008</year>
<volume>127</volume>
<number>1</number>
<pages>81–88</pages>
<issn>0025-2611</issn>
<mrnumber>2429915 (2009g:20149)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Ramiro Carrillo-Catalán</mrreviewer>
<url>http://dx.doi.org/10.1007/s00229-008-0188-5</url>
<other type="coden">MSMHB2</other>
<other type="doi">10.1007/s00229-008-0188-5</other>
<other type="fjournal">Manuscripta Mathematica</other>
</article></entry>
<entry id="NaVo2003"><article>
<author>
<name><first>Gábor P.</first><last>Nagy</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Octonions, simple <C>M</C>oufang loops and triality</title>
<journal>Quasigroups Related Systems</journal>
<year>2003</year>
<volume>10</volume>
<pages>65–94</pages>
<issn>1561-2848</issn>
<mrnumber>1998692 (2004f:20118)</mrnumber>
<mrclass>20N05 (17A75)</mrclass>
<mrreviewer>Orin Chein</mrreviewer>
<other type="fjournal">Quasigroups and Related Systems</other>
</article></entry>
<entry id="NaVo2007"><article>
<author>
<name><first>Gábor P.</first><last>Nagy</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>The <C>M</C>oufang loops of order 64 and 81</title>
<journal>J. Symbolic Comput.</journal>
<year>2007</year>
<volume>42</volume>
<number>9</number>
<pages>871–883</pages>
<issn>0747-7171</issn>
<mrnumber>2355056 (2009d:20155)</mrnumber>
<mrclass>20N05 (20D15)</mrclass>
<mrreviewer>Chris A. Rowley</mrreviewer>
<url>http://dx.doi.org/10.1016/j.jsc.2007.06.004</url>
<other type="doi">10.1016/j.jsc.2007.06.004</other>
<other type="fjournal">Journal of Symbolic Computation</other>
</article></entry>
<entry id="Pf"><book>
<author>
<name><first>Hala O.</first><last>Pflugfelder</last></name>
</author>
<title>Quasigroups and loops: introduction</title>
<publisher>Heldermann Verlag</publisher>
<year>1990</year>
<volume>7</volume>
<series>Sigma Series in Pure Mathematics</series>
<address>Berlin</address>
<isbn>3-88538-007-2</isbn>
<mrnumber>1125767 (93g:20132)</mrnumber>
<mrclass>20N05 (20-01)</mrclass>
<mrreviewer>D. A. Robinson</mrreviewer>
<other type="pages">viii+147</other>
</book></entry>
<entry id="PhiVoj"><article>
<author>
<name><first>J. D.</first><last>Phillips</last></name>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>The varieties of loops of <C>B</C>ol-<C>M</C>oufang type</title>
<journal>Algebra Universalis</journal>
<year>2005</year>
<volume>54</volume>
<number>3</number>
<pages>259–271</pages>
<issn>0002-5240</issn>
<mrnumber>2219409 (2007b:20147)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>A. Schleiermacher</mrreviewer>
<url>http://dx.doi.org/10.1007/s00012-005-1941-1</url>
<other type="coden">AGUVA3</other>
<other type="doi">10.1007/s00012-005-1941-1</other>
<other type="fjournal">Algebra Universalis</other>
</article></entry>
<entry id="SlZe2011"><article>
<author>
<name><first>M.</first><last>Slattery</last></name>
<name><first>A.</first><last>Zenisek</last></name>
</author>
<title>Moufang loops of order 243</title>
<journal>Commentationes Mathematicae Universitatis Carolinae</journal>
<year>2012</year>
<volume>53</volume>
<number>3</number>
<pages>423–428</pages>
</article></entry>
<entry id="Vo"><article>
<author>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Toward the classification of <C>M</C>oufang loops of order 64</title>
<journal>European J. Combin.</journal>
<year>2006</year>
<volume>27</volume>
<number>3</number>
<pages>444–460</pages>
<issn>0195-6698</issn>
<mrnumber>2206479 (2006k:20136)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Orin Chein</mrreviewer>
<url>http://dx.doi.org/10.1016/j.ejc.2004.10.003</url>
<other type="doi">10.1016/j.ejc.2004.10.003</other>
<other type="fjournal">European Journal of Combinatorics</other>
</article></entry>
<entry id="VoQRS"><article>
<author>
<name><first>Petr</first><last>Vojtěchovský</last></name>
</author>
<title>Three lectures on automorphic loops</title>
<journal>Quasigroups Related Systems</journal>
<year>2015</year>
<volume>23</volume>
<number>1</number>
<pages>129–163</pages>
<issn>1561-2848</issn>
<mrnumber>3353114</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>Ágota Figula</mrreviewer>
<other type="fjournal">Quasigroups and Related Systems</other>
</article></entry>
<entry id="Wi"><article>
<author>
<name><first>Robert L.</first><last>Wilson Jr. </last></name>
</author>
<title>Quasidirect products of quasigroups</title>
<journal>Comm. Algebra</journal>
<year>1975</year>
<volume>3</volume>
<number>9</number>
<pages>835–850</pages>
<issn>0092-7872</issn>
<mrnumber>0376937 (51 #13112)</mrnumber>
<mrclass>20N05</mrclass>
<mrreviewer>D. A. Robinson</mrreviewer>
<other type="fjournal">Communications in Algebra</other>
</article></entry>
</file>

496
doc/loops_bib.xml.bib
View File

@ -1,496 +0,0 @@
@phdthesis{ Artic,
author = {Artic, K.},
title = {On conjugacy closed loops and conjugacy closed loop
folders},
school = {RWTH Aachen University},
year = {2015},
printedkey = {Art15}
}
@article{ Ar,
author = {Artzy, R.},
title = {On automorphic-inverse properties in loops},
journal = {Proc. Amer. Math. Soc.},
volume = {10},
year = {1959},
pages = {588{\textendash}591},
fjournal = {Proceedings of the American Mathematical Society},
issn = {0002-9939},
mrclass = {20.00},
mrnumber = {0107674 (21 \#6397)},
mrreviewer = {H. Minc},
printedkey = {Art59}
}
@article{ BaGrVo,
author = {De Barros, D. A. S. and Grishkov, A. and Vojt{\v
e}chovsk{\a'y}, P.},
title = {Commutative automorphic loops of order {$p^3$}},
journal = {J. Algebra Appl.},
volume = {11},
number = {5},
year = {2012},
pages = {1250100, 15},
fjournal = {Journal of Algebra and its Applications},
issn = {0219-4988},
mrclass = {20N05 (20G40)},
mrnumber = {2983192},
mrreviewer = {{\a'A}gota Figula},
url = {http://dx.doi.org/10.1142/S0219498812501009},
doi = {10.1142/S0219498812501009},
printedkey = {BGV12}
}
@book{ Br,
author = {Bruck, R. H.},
title = {A survey of binary systems},
publisher = {Springer Verlag},
series = {Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue
Folge, Heft 20. Reihe: Gruppentheorie},
address = {Berlin},
year = {1958},
pages = {viii+185},
mrclass = {20.00},
mrnumber = {0093552 (20 \#76)},
mrreviewer = {L. J. Paige},
printedkey = {Bru58}
}
@article{ BrPa,
author = {Bruck, R. H. and Paige, L. J.},
title = {Loops whose inner mappings are automorphisms},
journal = {Ann. of Math. (2)},
volume = {63},
year = {1956},
pages = {308{\textendash}323},
fjournal = {Annals of Mathematics. Second Series},
issn = {0003-486X},
mrclass = {20.0X},
mrnumber = {0076779},
mrreviewer = {R. Moufang},
printedkey = {BP56}
}
@book{ ChPfSm,
editor = {Chein, O. and Pflugfelder, H. O. and Smith, J. D. H.},
title = {Quasigroups and loops: theory and applications},
publisher = {Heldermann Verlag},
series = {Sigma Series in Pure Mathematics},
volume = {8},
address = {Berlin},
year = {1990},
pages = {xii+568},
isbn = {3-88538-008-0},
mrclass = {20N05 (20-06)},
mrnumber = {1125806 (93g:20133)},
mrreviewer = {D. A. Robinson},
printedkey = {CPS90}
}
@book{ CoRo,
author = {Colbourn, C. J. and Rosa, A.},
title = {Triple systems},
publisher = {The Clarendon Press Oxford University Press},
series = {Oxford Mathematical Monographs},
address = {New York},
year = {1999},
pages = {xvi+560},
isbn = {0-19-853576-7},
mrclass = {05B07 (05-02)},
mrnumber = {1843379 (2002h:05024)},
mrreviewer = {Elizabeth J. Billington},
printedkey = {CR99}
}
@article{ CsDr,
author = {Cs{\"o}rg{\H o}, P. and Dr{\a'a}pal, A.},
title = {Left conjugacy closed loops of nilpotency class two},
journal = {Results Math.},
volume = {47},
number = {3-4},
year = {2005},
pages = {242{\textendash}265},
fjournal = {Results in Mathematics. Resultate der Mathematik},
issn = {1422-6383},
mrclass = {20N05},
mrnumber = {2153496 (2006b:20095)},
mrreviewer = {Huberta Lausch},
printedkey = {CD05}
}
@article{ DaVo,
author = {Daly, D. and Vojt{\v e}chovsk{\a'y}, P.},
title = {Enumeration of nilpotent loops via cohomology},
journal = {J. Algebra},
volume = {322},
number = {11},
year = {2009},
pages = {4080{\textendash}4098},
coden = {JALGA4},
fjournal = {Journal of Algebra},
issn = {0021-8693},
mrclass = {20N05 (20J99)},
mrnumber = {2556139 (2011e:20098)},
mrreviewer = {Yu. M. Movsisyan},
url = {http://dx.doi.org/10.1016/j.jalgebra.2009.03.042},
doi = {10.1016/j.jalgebra.2009.03.042},
printedkey = {DV09}
}
@article{ DrapalCD,
author = {Dr{\a'a}pal, A.},
title = {Cyclic and dihedral constructions of even order},
journal = {Comment. Math. Univ. Carolin.},
volume = {44},
number = {4},
year = {2003},
pages = {593{\textendash}614},
coden = {},
fjournal = {Commentationes Mathematicae Universitatis Carolinae},
issn = {},
mrclass = {20D60 (05B15)},
mrnumber = {MR2062876 (2005d:20038)},
mrreviewer = {Thomas Michael Keller},
url = {},
doi = {},
printedkey = {Dr{\a'a}03}
}
@article{ DrVo,
author = {Dr{\a'a}pal, A. and Vojt{\v e}chovsk{\a'y}, P.},
title = {Moufang loops that share associator and three quarters
of their multiplication tables},
journal = {Rocky Mountain J. Math.},
volume = {36},
number = {2},
year = {2006},
pages = {425{\textendash}455},
coden = {RMJMAE},
fjournal = {The Rocky Mountain Journal of Mathematics},
issn = {0035-7596},
mrclass = {20N05 (05B15)},
mrnumber = {2234814 (2007d:20114)},
mrreviewer = {Orin Chein},
url = {http://dx.doi.org/10.1216/rmjm/1181069461},
doi = {10.1216/rmjm/1181069461},
printedkey = {DV06}
}
@article{ Fe,
author = {Fenyves, F.},
title = {Extra loops. {II}. {O}n loops with identities of
{B}ol-{M}oufang type},
journal = {Publ. Math. Debrecen},
volume = {16},
year = {1969},
pages = {187{\textendash}192},
fjournal = {Publicationes Mathematicae Debrecen},
issn = {0033-3883},
mrclass = {20.95},
mrnumber = {0262409 (41 \#7017)},
mrreviewer = {D. A. Robinson},
printedkey = {Fen69}
}
@book{ Go,
author = {Goodaire, E. G. and May, S. and Raman, M.},
title = {The {M}oufang loops of order less than 64},
publisher = {Nova Science Publishers Inc.},
address = {Commack, NY},
year = {1999},
pages = {xviii+287},
isbn = {1-56072-659-8},
mrclass = {20N05},
mrnumber = {1689624 (2000a:20147)},
printedkey = {GMR99}
}
@article{ GrKiNa,
author = {Grishkov, A. and Kinyon, M. and Nagy, G. P.},
title = {Solvability of commutative automorphic loops},
journal = {Proc. Amer. Math. Soc.},
volume = {142},
number = {9},
year = {2014},
pages = {3029{\textendash}3037},
fjournal = {Proceedings of the American Mathematical Society},
issn = {0002-9939},
mrclass = {20N05 (17B99)},
mrnumber = {3223359},
mrreviewer = {J. D. Phillips},
url = {http://dx.doi.org/10.1090/S0002-9939-2014-12053-3},
doi = {10.1090/S0002-9939-2014-12053-3},
printedkey = {GKN14}
}
@article{ JaMa,
author = {Jacobson, M. T. and Matthews, P.},
title = {Generating uniformly distributed random {L}atin
squares},
journal = {J. Combin. Des.},
volume = {4},
number = {6},
year = {1996},
pages = {405{\textendash}437},
fjournal = {Journal of Combinatorial Designs},
issn = {1063-8539},
mrclass = {05B15 (60J10)},
mrnumber = {1410617 (98b:05021)},
mrreviewer = {Lars D{\o}vling Andersen},
url = {http://dx.doi.org/10.1002/(SICI)1520-6610(1996)4:6{\textless}405::AID-JCD3{\textgreater}3.0.CO;2-J},
doi = {10.1002/(SICI)1520-6610(1996)4:6{\textless}405::AID-JCD3{\textgreater}3.0.CO;2-J},
printedkey = {JM96}
}
@article{ JeKiVo,
author = {Jedli{\v c}ka, P. and Kinyon, M. and Vojt{\v
e}chovsk{\a'y}, P.},
title = {Nilpotency in automorphic loops of prime power order},
journal = {J. Algebra},
volume = {350},
year = {2012},
pages = {64{\textendash}76},
coden = {JALGA4},
fjournal = {Journal of Algebra},
issn = {0021-8693},
mrclass = {20N05},
mrnumber = {2859875},
mrreviewer = {Mohammad Shahryari},
url = {http://dx.doi.org/10.1016/j.jalgebra.2011.09.034},
doi = {10.1016/j.jalgebra.2011.09.034},
printedkey = {JKV12}
}
@article{ JoKiNaVo,
author = {Johnson, K. W. and Kinyon, M. K. and Nagy, G. P. and
Vojt{\v e}chovsk{\a'y}, P.},
title = {Searching for small simple automorphic loops},
journal = {LMS J. Comput. Math.},
volume = {14},
year = {2011},
pages = {200{\textendash}213},
fjournal = {LMS Journal of Computation and Mathematics},
issn = {1461-1570},
mrclass = {20N05 (20B15 20B40)},
mrnumber = {2831230},
mrreviewer = {Tuval S. Foguel},
url = {http://dx.doi.org/10.1112/S1461157010000173},
doi = {10.1112/S1461157010000173},
printedkey = {JKNV11}
}
@article{ KiKuPh,
author = {Kinyon, M. K. and Kunen, K. and Phillips, J. D.},
title = {Every diassociative {$A$}-loop is {M}oufang},
journal = {Proc. Amer. Math. Soc.},
volume = {130},
number = {3},
year = {2002},
pages = {619{\textendash}624},
coden = {PAMYAR},
fjournal = {Proceedings of the American Mathematical Society},
issn = {0002-9939},
mrclass = {20N05 (68T15)},
mrnumber = {1866009 (2002k:20124)},
mrreviewer = {Orin Chein},
url = {http://dx.doi.org/10.1090/S0002-9939-01-06090-7},
doi = {10.1090/S0002-9939-01-06090-7},
printedkey = {KKP02}
}
@article{ KiKuPhVo,
author = {Kinyon, M. K. and Kunen, K. and Phillips, J. D. and
Vojt{\v e}chovsk{\a'y}, P.},
title = {The structure of automorphic loops},
journal = {Trans. Amer. Math. Soc.},
volume = {368},
number = {12},
year = {2016},
pages = {8901{\textendash}8927},
coden = {TAMTAM},
fjournal = {Transactions of the American Mathematical Society},
issn = {0002-9947},
mrclass = {20N05},
mrnumber = {3551593},
url = {http://dx.doi.org/10.1090/tran/6622},
doi = {10.1090/tran/6622},
printedkey = {KKPV16}
}
@article{ KiNaVo2015,
author = {Kinyon, M. K. and Nagy, G. P. and Vojt{\v
e}chovsk{\a'y}, P.},
title = {Bol loops and Bruck loops of order $pq$},
journal = {},
year = {2015},
note = {preprint},
printedkey = {KNV15}
}
@article{ Kun,
author = {Kunen, K.},
title = {The structure of conjugacy closed loops},
journal = {Trans. Amer. Math. Soc.},
volume = {352},
number = {6},
year = {2000},
pages = {2889{\textendash}2911},
coden = {TAMTAM},
fjournal = {Transactions of the American Mathematical Society},
issn = {0002-9947},
mrclass = {20N05 (03C05)},
mrnumber = {1615991 (2000j:20132)},
mrreviewer = {Edgar G. Goodaire},
url = {http://dx.doi.org/10.1090/S0002-9947-00-02350-3},
doi = {10.1090/S0002-9947-00-02350-3},
printedkey = {Kun00}
}
@article{ Li,
author = {Liebeck, M. W.},
title = {The classification of finite simple {M}oufang loops},
journal = {Math. Proc. Cambridge Philos. Soc.},
volume = {102},
number = {1},
year = {1987},
pages = {33{\textendash}47},
coden = {MPCPCO},
fjournal = {Mathematical Proceedings of the Cambridge
Philosophical Society},
issn = {0305-0041},
mrclass = {20N05},
mrnumber = {886433 (88g:20146)},
mrreviewer = {Karl H. Robinson},
url = {http://dx.doi.org/10.1017/S0305004100067025},
doi = {10.1017/S0305004100067025},
printedkey = {Lie87}
}
@unpublished{ Mo,
author = {Moorhouse, G. E.},
title = {Bol loops of small order},
note = {http://www.uwyo.edu/moorhouse/pub/bol/},
printedkey = {Moo}
}
@article{ Na,
author = {Nagy, G. P.},
title = {A class of simple proper {B}ol loops},
journal = {Manuscripta Math.},
volume = {127},
number = {1},
year = {2008},
pages = {81{\textendash}88},
coden = {MSMHB2},
fjournal = {Manuscripta Mathematica},
issn = {0025-2611},
mrclass = {20N05},
mrnumber = {2429915 (2009g:20149)},
mrreviewer = {Ramiro Carrillo-Catal{\a'a}n},
url = {http://dx.doi.org/10.1007/s00229-008-0188-5},
doi = {10.1007/s00229-008-0188-5},
printedkey = {Nag08}
}
@article{ NaVo2003,
author = {Nagy, G. P. and Vojt{\v e}chovsk{\a'y}, P.},
title = {Octonions, simple {M}oufang loops and triality},
journal = {Quasigroups Related Systems},
volume = {10},
year = {2003},
pages = {65{\textendash}94},
fjournal = {Quasigroups and Related Systems},
issn = {1561-2848},
mrclass = {20N05 (17A75)},
mrnumber = {1998692 (2004f:20118)},
mrreviewer = {Orin Chein},
printedkey = {NV03}
}
@article{ NaVo2007,
author = {Nagy, G. P. and Vojt{\v e}chovsk{\a'y}, P.},
title = {The {M}oufang loops of order 64 and 81},
journal = {J. Symbolic Comput.},
volume = {42},
number = {9},
year = {2007},
pages = {871{\textendash}883},
fjournal = {Journal of Symbolic Computation},
issn = {0747-7171},
mrclass = {20N05 (20D15)},
mrnumber = {2355056 (2009d:20155)},
mrreviewer = {Chris A. Rowley},
url = {http://dx.doi.org/10.1016/j.jsc.2007.06.004},
doi = {10.1016/j.jsc.2007.06.004},
printedkey = {NV07}
}
@book{ Pf,
author = {Pflugfelder, H. O.},
title = {Quasigroups and loops: introduction},
publisher = {Heldermann Verlag},
series = {Sigma Series in Pure Mathematics},
volume = {7},
address = {Berlin},
year = {1990},
pages = {viii+147},
isbn = {3-88538-007-2},
mrclass = {20N05 (20-01)},
mrnumber = {1125767 (93g:20132)},
mrreviewer = {D. A. Robinson},
printedkey = {Pfl90}
}
@article{ PhiVoj,
author = {Phillips, J. D. and Vojt{\v e}chovsk{\a'y}, P.},
title = {The varieties of loops of {B}ol-{M}oufang type},
journal = {Algebra Universalis},
volume = {54},
number = {3},
year = {2005},
pages = {259{\textendash}271},
coden = {AGUVA3},
fjournal = {Algebra Universalis},
issn = {0002-5240},
mrclass = {20N05},
mrnumber = {2219409 (2007b:20147)},
mrreviewer = {A. Schleiermacher},
url = {http://dx.doi.org/10.1007/s00012-005-1941-1},
doi = {10.1007/s00012-005-1941-1},
printedkey = {PV05}
}
@article{ SlZe2011,
author = {Slattery, M. and Zenisek, A.},
title = {Moufang loops of order 243},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {53},
number = {3},
year = {2012},
pages = {423{\textendash}428},
printedkey = {SZ12}
}
@article{ Vo,
author = {Vojt{\v e}chovsk{\a'y}, P.},
title = {Toward the classification of {M}oufang loops of order
64},
journal = {European J. Combin.},
volume = {27},
number = {3},
year = {2006},
pages = {444{\textendash}460},
fjournal = {European Journal of Combinatorics},
issn = {0195-6698},
mrclass = {20N05},
mrnumber = {2206479 (2006k:20136)},
mrreviewer = {Orin Chein},
url = {http://dx.doi.org/10.1016/j.ejc.2004.10.003},
doi = {10.1016/j.ejc.2004.10.003},
printedkey = {Voj06}
}
@article{ VoQRS,
author = {Vojt{\v e}chovsk{\a'y}, P.},
title = {Three lectures on automorphic loops},
journal = {Quasigroups Related Systems},
volume = {23},
number = {1},
year = {2015},
pages = {129{\textendash}163},
fjournal = {Quasigroups and Related Systems},
issn = {1561-2848},
mrclass = {20N05},
mrnumber = {3353114},
mrreviewer = {{\a'A}gota Figula},
printedkey = {Voj15}
}
@article{ Wi,
author = {Wilson Jr., R. L.},
title = {Quasidirect products of quasigroups},
journal = {Comm. Algebra},
volume = {3},
number = {9},
year = {1975},
pages = {835{\textendash}850},
fjournal = {Communications in Algebra},
issn = {0092-7872},
mrclass = {20N05},
mrnumber = {0376937 (51 \#13112)},
mrreviewer = {D. A. Robinson},
printedkey = {WJ75}
}

BIN
doc/manual.dvi
View File

Binary file not shown.

19749
doc/manual.pdf
View File

File diff suppressed because it is too large Load Diff

9307
doc/manual.ps
View File

File diff suppressed because it is too large Load Diff

556
doc/manual.six
View File

@ -258,134 +258,138 @@ X", "6.7", [ 6, 7, 0 ], 320, 31, "normal subloops and simple loops",
, "6.11", [ 6, 11, 0 ], 444, 33, "isomorphisms and automorphisms",
"X81F3496578EAA74E" ],
[ "\033[1X\033[33X\033[0;-2YIsotopisms\033[133X\033[101X", "6.12",
[ 6, 12, 0 ], 532, 35, "isotopisms", "X7E996BDD81E594F9" ],
[ 6, 12, 0 ], 543, 35, "isotopisms", "X7E996BDD81E594F9" ],
[
"\033[1X\033[33X\033[0;-2YTesting Properties of Quasigroups and Loops\033[1\
33X\033[101X", "7", [ 7, 0, 0 ], 1, 36,
33X\033[101X", "7", [ 7, 0, 0 ], 1, 37,
"testing properties of quasigroups and loops", "X7910E575825C713E" ],
[
"\033[1X\033[33X\033[0;-2YAssociativity, Commutativity and Generalizations\\
033[133X\033[101X", "7.1", [ 7, 1, 0 ], 16, 36,
033[133X\033[101X", "7.1", [ 7, 1, 0 ], 16, 37,
"associativity commutativity and generalizations", "X7960E3FB7A7F0F00" ]
, [ "\033[1X\033[33X\033[0;-2YInverse Propeties\033[133X\033[101X",
"7.2", [ 7, 2, 0 ], 46, 37, "inverse propeties", "X853841C5820BFEA4" ],
"7.2", [ 7, 2, 0 ], 46, 38, "inverse propeties", "X853841C5820BFEA4" ],
[ "\033[1X\033[33X\033[0;-2YHasLeftInverseProperty, HasRightInverseProperty \
and HasInverseProperty\033[133X\033[101X", "7.2-1", [ 7, 2, 1 ], 53, 37,
and HasInverseProperty\033[133X\033[101X", "7.2-1", [ 7, 2, 1 ], 53, 38,
"hasleftinverseproperty hasrightinverseproperty and hasinverseproperty",
"X85EDD10586596458" ],
[
"\033[1X\033[33X\033[0;-2YSome Properties of Quasigroups\033[133X\033[101X"
, "7.3", [ 7, 3, 0 ], 102, 38, "some properties of quasigroups",
, "7.3", [ 7, 3, 0 ], 102, 39, "some properties of quasigroups",
"X7D8CB6DA828FD744" ],
[
"\033[1X\033[33X\033[0;-2YIsLeftDistributive, IsRightDistributive, IsDistri\
butive\033[133X\033[101X", "7.3-6", [ 7, 3, 6 ], 143, 38,
butive\033[133X\033[101X", "7.3-6", [ 7, 3, 6 ], 143, 39,
"isleftdistributive isrightdistributive isdistributive",
"X7B76FD6E878ED4F1" ],
[ "\033[1X\033[33X\033[0;-2YIsEntropic and IsMedial\033[133X\033[101X",
"7.3-7", [ 7, 3, 7 ], 160, 39, "isentropic and ismedial",
"7.3-7", [ 7, 3, 7 ], 160, 40, "isentropic and ismedial",
"X7F23D4D97A38D223" ],
[ "\033[1X\033[33X\033[0;-2YLoops of Bol Moufang Type\033[133X\033[101X",
"7.4", [ 7, 4, 0 ], 170, 39, "loops of bol moufang type",
"7.4", [ 7, 4, 0 ], 170, 40, "loops of bol moufang type",
"X780D907986EBA6C7" ],
[ "\033[1X\033[33X\033[0;-2YPower Alternative Loops\033[133X\033[101X",
"7.5", [ 7, 5, 0 ], 324, 42, "power alternative loops",
"7.5", [ 7, 5, 0 ], 324, 43, "power alternative loops",
"X83A501387E1AC371" ],
[
"\033[1X\033[33X\033[0;-2YIsLeftPowerAlternative, IsRightPowerAlternative a\
nd IsPowerAlternative\033[133X\033[101X", "7.5-1", [ 7, 5, 1 ], 337, 42,
nd IsPowerAlternative\033[133X\033[101X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"isleftpoweralternative isrightpoweralternative and ispoweralternative",
"X875C3DF681B3FAE2" ],
[
"\033[1X\033[33X\033[0;-2YConjugacy Closed Loops and Related Properties\\
033[133X\033[101X", "7.6", [ 7, 6, 0 ], 346, 42,
033[133X\033[101X", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loops and related properties", "X8176B2C47A4629CD" ],
[ "\033[1X\033[33X\033[0;-2YAutomorphic Loops\033[133X\033[101X", "7.7",
[ 7, 7, 0 ], 384, 43, "automorphic loops", "X793B22EA8643C667" ],
[ 7, 7, 0 ], 384, 44, "automorphic loops", "X793B22EA8643C667" ],
[ "\033[1X\033[33X\033[0;-2YAdditonal Varieties of Loops\033[133X\033[101X",
"7.8", [ 7, 8, 0 ], 451, 44, "additonal varieties of loops",
"7.8", [ 7, 8, 0 ], 451, 45, "additonal varieties of loops",
"X846F363879BAB349" ],
[
"\033[1X\033[33X\033[0;-2YIsLeftBruckLoop and IsLeftKLoop\033[133X\033[101X\
", "7.8-3", [ 7, 8, 3 ], 470, 44, "isleftbruckloop and isleftkloop",
", "7.8-3", [ 7, 8, 3 ], 470, 45, "isleftbruckloop and isleftkloop",
"X85F1BD4280E44F5B" ],
[
"\033[1X\033[33X\033[0;-2YIsRightBruckLoop and IsRightKLoop\033[133X\033[10\
1X", "7.8-4", [ 7, 8, 4 ], 480, 44, "isrightbruckloop and isrightkloop",
1X", "7.8-4", [ 7, 8, 4 ], 480, 45, "isrightbruckloop and isrightkloop",
"X857B373E7B4E0519" ],
[ "\033[1X\033[33X\033[0;-2YSpecific Methods\033[133X\033[101X", "8",
[ 8, 0, 0 ], 1, 45, "specific methods", "X85AFC9C47FD3C03F" ],
[ 8, 0, 0 ], 1, 46, "specific methods", "X85AFC9C47FD3C03F" ],
[ "\033[1X\033[33X\033[0;-2YCore Methods for Bol Loops\033[133X\033[101X",
"8.1", [ 8, 1, 0 ], 7, 45, "core methods for bol loops",
"8.1", [ 8, 1, 0 ], 7, 46, "core methods for bol loops",
"X7990F2F880E717EE" ],
[
"\033[1X\033[33X\033[0;-2YAssociatedLeftBruckLoop and AssociatedRightBruckL\
oop\033[133X\033[101X", "8.1-1", [ 8, 1, 1 ], 10, 45,
oop\033[133X\033[101X", "8.1-1", [ 8, 1, 1 ], 10, 46,
"associatedleftbruckloop and associatedrightbruckloop",
"X8664CA927DD73DBE" ],
[ "\033[1X\033[33X\033[0;-2YMoufang Modifications\033[133X\033[101X",
"8.2", [ 8, 2, 0 ], 47, 46, "moufang modifications",
"8.2", [ 8, 2, 0 ], 47, 47, "moufang modifications",
"X819F82737C2A860D" ],
[ "\033[1X\033[33X\033[0;-2YTriality for Moufang Loops\033[133X\033[101X",
"8.3", [ 8, 3, 0 ], 98, 46, "triality for moufang loops",
"8.3", [ 8, 3, 0 ], 98, 47, "triality for moufang loops",
"X83E73A767D79FAFD" ],
[
"\033[1X\033[33X\033[0;-2YRealizing Groups as Multiplication Groups of Loop\
s\033[133X\033[101X", "8.4", [ 8, 4, 0 ], 127, 47,
s\033[133X\033[101X", "8.4", [ 8, 4, 0 ], 127, 48,
"realizing groups as multiplication groups of loops",
"X841ED66B8084AA73" ],
[ "\033[1X\033[33X\033[0;-2YLibraries of Loops\033[133X\033[101X", "9",
[ 9, 0, 0 ], 1, 49, "libraries of loops", "X7BF3EE6E7953560D" ],
[ 9, 0, 0 ], 1, 50, "libraries of loops", "X7BF3EE6E7953560D" ],
[ "\033[1X\033[33X\033[0;-2YA Typical Library\033[133X\033[101X", "9.1",
[ 9, 1, 0 ], 7, 49, "a typical library", "X874DFEAA79B3377C" ],
[ 9, 1, 0 ], 7, 50, "a typical library", "X874DFEAA79B3377C" ],
[
"\033[1X\033[33X\033[0;-2YLeft Bol Loops and Right Bol Loops\033[133X\033[1\
01X", "9.2", [ 9, 2, 0 ], 54, 50, "left bol loops and right bol loops",
01X", "9.2", [ 9, 2, 0 ], 54, 51, "left bol loops and right bol loops",
"X7DF21BD685FBF258" ],
[ "\033[1X\033[33X\033[0;-2YMoufang Loops\033[133X\033[101X", "9.3",
[ 9, 3, 0 ], 80, 50, "moufang loops", "X7953702D84E60AF4" ],
[ "\033[1X\033[33X\033[0;-2YCode Loops\033[133X\033[101X", "9.4",
[ 9, 4, 0 ], 109, 51, "code loops", "X7BCA6BCB847F79DC" ],
[ "\033[1X\033[33X\033[0;-2YSteiner Loops\033[133X\033[101X", "9.5",
[ 9, 5, 0 ], 122, 51, "steiner loops", "X84E941EE7846D3EE" ],
[
"\033[1X\033[33X\033[0;-2YLeft Bruck Loops and Right Bruck Loops\033[133X\\
033[101X", "9.3", [ 9, 3, 0 ], 80, 51,
"left bruck loops and right bruck loops", "X8028D69A86B15897" ],
[ "\033[1X\033[33X\033[0;-2YMoufang Loops\033[133X\033[101X", "9.4",
[ 9, 4, 0 ], 102, 52, "moufang loops", "X7953702D84E60AF4" ],
[ "\033[1X\033[33X\033[0;-2YCode Loops\033[133X\033[101X", "9.5",
[ 9, 5, 0 ], 131, 52, "code loops", "X7BCA6BCB847F79DC" ],
[ "\033[1X\033[33X\033[0;-2YSteiner Loops\033[133X\033[101X", "9.6",
[ 9, 6, 0 ], 144, 52, "steiner loops", "X84E941EE7846D3EE" ],
[ "\033[1X\033[33X\033[0;-2YConjugacy Closed Loops\033[133X\033[101X",
"9.6", [ 9, 6, 0 ], 149, 51, "conjugacy closed loops",
"9.7", [ 9, 7, 0 ], 171, 53, "conjugacy closed loops",
"X867E5F0783FEB8B5" ],
[
"\033[1X\033[33X\033[0;-2YRCCLoop and RightConjugacyClosedLoop\033[133X\\
033[101X", "9.6-1", [ 9, 6, 1 ], 173, 52,
033[101X", "9.7-1", [ 9, 7, 1 ], 195, 53,
"rccloop and rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[
"\033[1X\033[33X\033[0;-2YLCCLoop and LeftConjugacyClosedLoop\033[133X\033[\
101X", "9.6-2", [ 9, 6, 2 ], 180, 52, "lccloop and leftconjugacyclosedloop",
101X", "9.7-2", [ 9, 7, 2 ], 202, 53, "lccloop and leftconjugacyclosedloop",
"X80AB8B107D55FB19" ],
[
"\033[1X\033[33X\033[0;-2YCCLoop and ConjugacyClosedLoop\033[133X\033[101X"
, "9.6-3", [ 9, 6, 3 ], 217, 52, "ccloop and conjugacyclosedloop",
, "9.7-3", [ 9, 7, 3 ], 241, 54, "ccloop and conjugacyclosedloop",
"X798BC601843E8916" ],
[ "\033[1X\033[33X\033[0;-2YSmall Loops\033[133X\033[101X", "9.7",
[ 9, 7, 0 ], 224, 52, "small loops", "X7E3A8F2C790F2CA1" ],
[ "\033[1X\033[33X\033[0;-2YPaige Loops\033[133X\033[101X", "9.8",
[ 9, 8, 0 ], 235, 53, "paige loops", "X8135C8FD8714C606" ],
[ "\033[1X\033[33X\033[0;-2YNilpotent Loops\033[133X\033[101X", "9.9",
[ 9, 9, 0 ], 250, 53, "nilpotent loops", "X86695C577A4D1784" ],
[ "\033[1X\033[33X\033[0;-2YAutomorphic Loops\033[133X\033[101X", "9.10",
[ 9, 10, 0 ], 266, 53, "automorphic loops", "X793B22EA8643C667" ],
[ "\033[1X\033[33X\033[0;-2YInteresting Loops\033[133X\033[101X", "9.11",
[ 9, 11, 0 ], 283, 54, "interesting loops", "X843BD73F788049F7" ],
[ "\033[1X\033[33X\033[0;-2YSmall Loops\033[133X\033[101X", "9.8",
[ 9, 8, 0 ], 248, 54, "small loops", "X7E3A8F2C790F2CA1" ],
[ "\033[1X\033[33X\033[0;-2YPaige Loops\033[133X\033[101X", "9.9",
[ 9, 9, 0 ], 259, 54, "paige loops", "X8135C8FD8714C606" ],
[ "\033[1X\033[33X\033[0;-2YNilpotent Loops\033[133X\033[101X", "9.10",
[ 9, 10, 0 ], 274, 54, "nilpotent loops", "X86695C577A4D1784" ],
[ "\033[1X\033[33X\033[0;-2YAutomorphic Loops\033[133X\033[101X", "9.11",
[ 9, 11, 0 ], 290, 55, "automorphic loops", "X793B22EA8643C667" ],
[ "\033[1X\033[33X\033[0;-2YInteresting Loops\033[133X\033[101X", "9.12",
[ 9, 12, 0 ], 309, 55, "interesting loops", "X843BD73F788049F7" ],
[
"\033[1X\033[33X\033[0;-2YLibraries of Loops Up To Isotopism\033[133X\033[1\
01X", "9.12", [ 9, 12, 0 ], 298, 54, "libraries of loops up to isotopism",
01X", "9.13", [ 9, 13, 0 ], 324, 55, "libraries of loops up to isotopism",
"X864839227D5C0A90" ],
[ "\033[1X\033[33X\033[0;-2YFiles\033[133X\033[101X", "a", [ "A", 0, 0 ],
1, 55, "files", "X7BC4571A79FFB7D0" ],
1, 56, "files", "X7BC4571A79FFB7D0" ],
[ "\033[1X\033[33X\033[0;-2YFilters\033[133X\033[101X", "b", [ "B", 0, 0 ],
1, 57, "filters", "X84EFA4C07D4277BB" ],
[ "Bibliography", "bib", [ "Bib", 0, 0 ], 1, 60, "bibliography",
1, 58, "filters", "X84EFA4C07D4277BB" ],
[ "Bibliography", "bib", [ "Bib", 0, 0 ], 1, 61, "bibliography",
"X7A6F98FD85F02BFE" ],
[ "References", "bib", [ "Bib", 0, 0 ], 1, 60, "references",
[ "References", "bib", [ "Bib", 0, 0 ], 1, 61, "references",
"X7A6F98FD85F02BFE" ],
[ "Index", "ind", [ "Ind", 0, 0 ], 1, 62, "index", "X83A0356F839C696F" ],
[ "Index", "ind", [ "Ind", 0, 0 ], 1, 63, "index", "X83A0356F839C696F" ],
[ "groupoid", "2.1", [ 2, 1, 0 ], 11, 8, "groupoid", "X80243DE5826583B8" ],
[ "magma", "2.1", [ 2, 1, 0 ], 11, 8, "magma", "X80243DE5826583B8" ],
[ "neutral element", "2.1", [ 2, 1, 0 ], 11, 8, "neutral element",
@ -790,438 +794,446 @@ s\033[133X\033[101X", "8.4", [ 8, 4, 0 ], 127, 47,
"loopsuptoisomorphism", "X8308F38283C61B20" ],
[ "\033[2XAutomorphismGroup\033[102X", "6.11-5", [ 6, 11, 5 ], 477, 34,
"automorphismgroup", "X87677B0787B4461A" ],
[ "\033[2XIsomorphicCopyByPerm\033[102X", "6.11-6", [ 6, 11, 6 ], 491, 34,
[ "\033[2XQuasigroupIsomorph\033[102X", "6.11-6", [ 6, 11, 6 ], 491, 34,
"quasigroupisomorph", "X7A42812B7B027DD4" ],
[ "\033[2XLoopIsomorph\033[102X", "6.11-7", [ 6, 11, 7 ], 498, 34,
"loopisomorph", "X7BD1AC32851286EA" ],
[ "\033[2XIsomorphicCopyByPerm\033[102X", "6.11-8", [ 6, 11, 8 ], 506, 35,
"isomorphiccopybyperm", "X85B3E22679FD8D81" ],
[ "\033[2XIsomorphicCopyByNormalSubloop\033[102X", "6.11-7", [ 6, 11, 7 ],
501, 34, "isomorphiccopybynormalsubloop", "X8121DE3A78795040" ],
[ "\033[2XDiscriminator\033[102X", "6.11-8", [ 6, 11, 8 ], 514, 35,
[ "\033[2XIsomorphicCopyByNormalSubloop\033[102X", "6.11-9", [ 6, 11, 9 ],
512, 35, "isomorphiccopybynormalsubloop", "X8121DE3A78795040" ],
[ "\033[2XDiscriminator\033[102X", "6.11-10", [ 6, 11, 10 ], 525, 35,
"discriminator", "X7D09D8957E4A0973" ],
[ "\033[2XAreEqualDiscriminators\033[102X", "6.11-9", [ 6, 11, 9 ], 526,
[ "\033[2XAreEqualDiscriminators\033[102X", "6.11-11", [ 6, 11, 11 ], 537,
35, "areequaldiscriminators", "X812F0DEE7C896E18" ],
[ "\033[2XIsotopismLoops\033[102X", "6.12-1", [ 6, 12, 1 ], 542, 35,
[ "\033[2XIsotopismLoops\033[102X", "6.12-1", [ 6, 12, 1 ], 553, 35,
"isotopismloops", "X84C5ADE77F910F63" ],
[ "\033[2XLoopsUpToIsotopism\033[102X", "6.12-2", [ 6, 12, 2 ], 548, 35,
[ "\033[2XLoopsUpToIsotopism\033[102X", "6.12-2", [ 6, 12, 2 ], 559, 36,
"loopsuptoisotopism", "X841E540B7A7EF29F" ],
[ "\033[2XIsAssociative\033[102X", "7.1-1", [ 7, 1, 1 ], 19, 36,
[ "\033[2XIsAssociative\033[102X", "7.1-1", [ 7, 1, 1 ], 19, 37,
"isassociative", "X7C83B5A47FD18FB7" ],
[ "\033[2XIsCommutative\033[102X", "7.1-2", [ 7, 1, 2 ], 24, 36,
[ "\033[2XIsCommutative\033[102X", "7.1-2", [ 7, 1, 2 ], 24, 37,
"iscommutative", "X830A4A4C795FBC2D" ],
[ "\033[2XIsPowerAssociative\033[102X", "7.1-3", [ 7, 1, 3 ], 29, 36,
[ "\033[2XIsPowerAssociative\033[102X", "7.1-3", [ 7, 1, 3 ], 29, 37,
"ispowerassociative", "X7D53EA947F1CDA69" ],
[ "quasigroup power associative", "7.1-3", [ 7, 1, 3 ], 29, 36,
[ "quasigroup power associative", "7.1-3", [ 7, 1, 3 ], 29, 37,
"quasigroup power associative", "X7D53EA947F1CDA69" ],
[ "power associative quasigroup", "7.1-3", [ 7, 1, 3 ], 29, 36,
[ "power associative quasigroup", "7.1-3", [ 7, 1, 3 ], 29, 37,
"power associative quasigroup", "X7D53EA947F1CDA69" ],
[ "\033[2XIsDiassociative\033[102X", "7.1-4", [ 7, 1, 4 ], 37, 36,
[ "\033[2XIsDiassociative\033[102X", "7.1-4", [ 7, 1, 4 ], 37, 37,
"isdiassociative", "X872DCA027E1A4A1D" ],
[ "quasigroup diassociative", "7.1-4", [ 7, 1, 4 ], 37, 36,
[ "quasigroup diassociative", "7.1-4", [ 7, 1, 4 ], 37, 37,
"quasigroup diassociative", "X872DCA027E1A4A1D" ],
[ "diassociative quasigroup", "7.1-4", [ 7, 1, 4 ], 37, 36,
[ "diassociative quasigroup", "7.1-4", [ 7, 1, 4 ], 37, 37,
"diassociative quasigroup", "X872DCA027E1A4A1D" ],
[ "inverse left", "7.2", [ 7, 2, 0 ], 46, 37, "inverse left",
[ "inverse left", "7.2", [ 7, 2, 0 ], 46, 38, "inverse left",
"X853841C5820BFEA4" ],
[ "inverse right", "7.2", [ 7, 2, 0 ], 46, 37, "inverse right",
[ "inverse right", "7.2", [ 7, 2, 0 ], 46, 38, "inverse right",
"X853841C5820BFEA4" ],
[ "\033[2XHasLeftInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 37,
[ "\033[2XHasLeftInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 38,
"hasleftinverseproperty", "X85EDD10586596458" ],
[ "\033[2XHasRightInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 37,
[ "\033[2XHasRightInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 38,
"hasrightinverseproperty", "X85EDD10586596458" ],
[ "\033[2XHasInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 37,
[ "\033[2XHasInverseProperty\033[102X", "7.2-1", [ 7, 2, 1 ], 53, 38,
"hasinverseproperty", "X85EDD10586596458" ],
[ "inverse property left", "7.2-1", [ 7, 2, 1 ], 53, 37,
[ "inverse property left", "7.2-1", [ 7, 2, 1 ], 53, 38,
"inverse property left", "X85EDD10586596458" ],
[ "inverse property right", "7.2-1", [ 7, 2, 1 ], 53, 37,
[ "inverse property right", "7.2-1", [ 7, 2, 1 ], 53, 38,
"inverse property right", "X85EDD10586596458" ],
[ "inverse property", "7.2-1", [ 7, 2, 1 ], 53, 37, "inverse property",
[ "inverse property", "7.2-1", [ 7, 2, 1 ], 53, 38, "inverse property",
"X85EDD10586596458" ],
[ "\033[2XHasTwosidedInverses\033[102X", "7.2-2", [ 7, 2, 2 ], 67, 37,
[ "\033[2XHasTwosidedInverses\033[102X", "7.2-2", [ 7, 2, 2 ], 67, 38,
"hastwosidedinverses", "X86B93E1B7AEA6EDA" ],
[ "inverse two-sided", "7.2-2", [ 7, 2, 2 ], 67, 37, "inverse two-sided",
[ "inverse two-sided", "7.2-2", [ 7, 2, 2 ], 67, 38, "inverse two-sided",
"X86B93E1B7AEA6EDA" ],
[ "\033[2XHasWeakInverseProperty\033[102X", "7.2-3", [ 7, 2, 3 ], 74, 37,
[ "\033[2XHasWeakInverseProperty\033[102X", "7.2-3", [ 7, 2, 3 ], 74, 38,
"hasweakinverseproperty", "X793909B780761EA8" ],
[ "inverse property weak", "7.2-3", [ 7, 2, 3 ], 74, 37,
[ "inverse property weak", "7.2-3", [ 7, 2, 3 ], 74, 38,
"inverse property weak", "X793909B780761EA8" ],
[ "\033[2XHasAutomorphicInverseProperty\033[102X", "7.2-4", [ 7, 2, 4 ],
82, 37, "hasautomorphicinverseproperty", "X7F46CE6B7D387158" ],
[ "automorphic inverse property", "7.2-4", [ 7, 2, 4 ], 82, 37,
82, 38, "hasautomorphicinverseproperty", "X7F46CE6B7D387158" ],
[ "automorphic inverse property", "7.2-4", [ 7, 2, 4 ], 82, 38,
"automorphic inverse property", "X7F46CE6B7D387158" ],
[ "inverse property automorphic", "7.2-4", [ 7, 2, 4 ], 82, 37,
[ "inverse property automorphic", "7.2-4", [ 7, 2, 4 ], 82, 38,
"inverse property automorphic", "X7F46CE6B7D387158" ],
[ "\033[2XHasAntiautomorphicInverseProperty\033[102X", "7.2-5",
[ 7, 2, 5 ], 91, 37, "hasantiautomorphicinverseproperty",
[ 7, 2, 5 ], 91, 38, "hasantiautomorphicinverseproperty",
"X8538D4638232DB51" ],
[ "antiautomorphic inverse property", "7.2-5", [ 7, 2, 5 ], 91, 37,
[ "antiautomorphic inverse property", "7.2-5", [ 7, 2, 5 ], 91, 38,
"antiautomorphic inverse property", "X8538D4638232DB51" ],
[ "inverse property antiautomorphic", "7.2-5", [ 7, 2, 5 ], 91, 37,
[ "inverse property antiautomorphic", "7.2-5", [ 7, 2, 5 ], 91, 38,
"inverse property antiautomorphic", "X8538D4638232DB51" ],
[ "\033[2XIsSemisymmetric\033[102X", "7.3-1", [ 7, 3, 1 ], 105, 38,
[ "\033[2XIsSemisymmetric\033[102X", "7.3-1", [ 7, 3, 1 ], 105, 39,
"issemisymmetric", "X834848ED85F9012B" ],
[ "semisymmetric quasigroup", "7.3-1", [ 7, 3, 1 ], 105, 38,
[ "semisymmetric quasigroup", "7.3-1", [ 7, 3, 1 ], 105, 39,
"semisymmetric quasigroup", "X834848ED85F9012B" ],
[ "quasigroup semisymmetric", "7.3-1", [ 7, 3, 1 ], 105, 38,
[ "quasigroup semisymmetric", "7.3-1", [ 7, 3, 1 ], 105, 39,
"quasigroup semisymmetric", "X834848ED85F9012B" ],
[ "\033[2XIsTotallySymmetric\033[102X", "7.3-2", [ 7, 3, 2 ], 113, 38,
[ "\033[2XIsTotallySymmetric\033[102X", "7.3-2", [ 7, 3, 2 ], 113, 39,
"istotallysymmetric", "X834F809B8060B754" ],
[ "totally symmetric quasigroup", "7.3-2", [ 7, 3, 2 ], 113, 38,
[ "totally symmetric quasigroup", "7.3-2", [ 7, 3, 2 ], 113, 39,
"totally symmetric quasigroup", "X834F809B8060B754" ],
[ "quasigroup totally symmetric", "7.3-2", [ 7, 3, 2 ], 113, 38,
[ "quasigroup totally symmetric", "7.3-2", [ 7, 3, 2 ], 113, 39,
"quasigroup totally symmetric", "X834F809B8060B754" ],
[ "\033[2XIsIdempotent\033[102X", "7.3-3", [ 7, 3, 3 ], 122, 38,
[ "\033[2XIsIdempotent\033[102X", "7.3-3", [ 7, 3, 3 ], 122, 39,
"isidempotent", "X7CB5896082D29173" ],
[ "idempotent quasigroup", "7.3-3", [ 7, 3, 3 ], 122, 38,
[ "idempotent quasigroup", "7.3-3", [ 7, 3, 3 ], 122, 39,
"idempotent quasigroup", "X7CB5896082D29173" ],
[ "quasigroup idempotent", "7.3-3", [ 7, 3, 3 ], 122, 38,
[ "quasigroup idempotent", "7.3-3", [ 7, 3, 3 ], 122, 39,
"quasigroup idempotent", "X7CB5896082D29173" ],
[ "\033[2XIsSteinerQuasigroup\033[102X", "7.3-4", [ 7, 3, 4 ], 129, 38,
[ "\033[2XIsSteinerQuasigroup\033[102X", "7.3-4", [ 7, 3, 4 ], 129, 39,
"issteinerquasigroup", "X83DE7DD77C056C1F" ],
[ "Steiner quasigroup", "7.3-4", [ 7, 3, 4 ], 129, 38, "steiner quasigroup",
[ "Steiner quasigroup", "7.3-4", [ 7, 3, 4 ], 129, 39, "steiner quasigroup",
"X83DE7DD77C056C1F" ],
[ "quasigroup Steiner", "7.3-4", [ 7, 3, 4 ], 129, 38, "quasigroup steiner",
[ "quasigroup Steiner", "7.3-4", [ 7, 3, 4 ], 129, 39, "quasigroup steiner",
"X83DE7DD77C056C1F" ],
[ "unipotent quasigroup", "7.3-5", [ 7, 3, 5 ], 136, 38,
[ "unipotent quasigroup", "7.3-5", [ 7, 3, 5 ], 136, 39,
"unipotent quasigroup", "X7CA3DCA07B6CB9BD" ],
[ "quasigroup unipotent", "7.3-5", [ 7, 3, 5 ], 136, 38,
[ "quasigroup unipotent", "7.3-5", [ 7, 3, 5 ], 136, 39,
"quasigroup unipotent", "X7CA3DCA07B6CB9BD" ],
[ "\033[2XIsUnipotent\033[102X", "7.3-5", [ 7, 3, 5 ], 136, 38,
[ "\033[2XIsUnipotent\033[102X", "7.3-5", [ 7, 3, 5 ], 136, 39,
"isunipotent", "X7CA3DCA07B6CB9BD" ],
[ "\033[2XIsLeftDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "\033[2XIsLeftDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 39,
"isleftdistributive", "X7B76FD6E878ED4F1" ],
[ "\033[2XIsRightDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "\033[2XIsRightDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 39,
"isrightdistributive", "X7B76FD6E878ED4F1" ],
[ "\033[2XIsDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "\033[2XIsDistributive\033[102X", "7.3-6", [ 7, 3, 6 ], 143, 39,
"isdistributive", "X7B76FD6E878ED4F1" ],
[ "quasigroup left distributive", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "quasigroup left distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup left distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup left", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup left", "X7B76FD6E878ED4F1" ],
[ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "quasigroup right distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup right distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup right", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup right", "X7B76FD6E878ED4F1" ],
[ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "quasigroup distributive", "7.3-6", [ 7, 3, 6 ], 143, 39,
"quasigroup distributive", "X7B76FD6E878ED4F1" ],
[ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 38,
[ "distributive quasigroup", "7.3-6", [ 7, 3, 6 ], 143, 39,
"distributive quasigroup", "X7B76FD6E878ED4F1" ],
[ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 39,
[ "\033[2XIsEntropic\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 40,
"isentropic", "X7F23D4D97A38D223" ],
[ "\033[2XIsMedial\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 39, "ismedial",
[ "\033[2XIsMedial\033[102X", "7.3-7", [ 7, 3, 7 ], 160, 40, "ismedial",
"X7F23D4D97A38D223" ],
[ "entropic quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 39,
[ "entropic quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 40,
"entropic quasigroup", "X7F23D4D97A38D223" ],
[ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 39,
[ "quasigroup entropic", "7.3-7", [ 7, 3, 7 ], 160, 40,
"quasigroup entropic", "X7F23D4D97A38D223" ],
[ "medial quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 39, "medial quasigroup",
[ "medial quasigroup", "7.3-7", [ 7, 3, 7 ], 160, 40, "medial quasigroup",
"X7F23D4D97A38D223" ],
[ "quasigroup medial", "7.3-7", [ 7, 3, 7 ], 160, 39, "quasigroup medial",
[ "quasigroup medial", "7.3-7", [ 7, 3, 7 ], 160, 40, "quasigroup medial",
"X7F23D4D97A38D223" ],
[ "loop of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 40,
"loop of bol-moufang type", "X780D907986EBA6C7" ],
[ "identity of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 39,
[ "identity of Bol-Moufang type", "7.4", [ 7, 4, 0 ], 170, 40,
"identity of bol-moufang type", "X780D907986EBA6C7" ],
[ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 39,
[ "alternative loop left", "7.4", [ 7, 4, 0 ], 170, 40,
"alternative loop left", "X780D907986EBA6C7" ],
[ "loop left alternative", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop left alternative", "7.4", [ 7, 4, 0 ], 170, 40,
"loop left alternative", "X780D907986EBA6C7" ],
[ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 39,
[ "alternative loop right", "7.4", [ 7, 4, 0 ], 170, 40,
"alternative loop right", "X780D907986EBA6C7" ],
[ "loop right alternative", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop right alternative", "7.4", [ 7, 4, 0 ], 170, 40,
"loop right alternative", "X780D907986EBA6C7" ],
[ "nuclear square loop left", "7.4", [ 7, 4, 0 ], 170, 39,
[ "nuclear square loop left", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop left", "X780D907986EBA6C7" ],
[ "loop left nuclear square", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop left nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop left nuclear square", "X780D907986EBA6C7" ],
[ "nuclear square loop middle", "7.4", [ 7, 4, 0 ], 170, 39,
[ "nuclear square loop middle", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop middle", "X780D907986EBA6C7" ],
[ "loop middle nuclear square", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop middle nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop middle nuclear square", "X780D907986EBA6C7" ],
[ "nuclear square loop right", "7.4", [ 7, 4, 0 ], 170, 39,
[ "nuclear square loop right", "7.4", [ 7, 4, 0 ], 170, 40,
"nuclear square loop right", "X780D907986EBA6C7" ],
[ "loop right nuclear square", "7.4", [ 7, 4, 0 ], 170, 39,
[ "loop right nuclear square", "7.4", [ 7, 4, 0 ], 170, 40,
"loop right nuclear square", "X780D907986EBA6C7" ],
[ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 39, "flexible loop",
[ "flexible loop", "7.4", [ 7, 4, 0 ], 170, 40, "flexible loop",
"X780D907986EBA6C7" ],
[ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 39, "loop flexible",
[ "loop flexible", "7.4", [ 7, 4, 0 ], 170, 40, "loop flexible",
"X780D907986EBA6C7" ],
[ "Bol loop left", "7.4", [ 7, 4, 0 ], 170, 39, "bol loop left",
[ "Bol loop left", "7.4", [ 7, 4, 0 ], 170, 40, "bol loop left",
"X780D907986EBA6C7" ],
[ "loop left Bol", "7.4", [ 7, 4, 0 ], 170, 39, "loop left bol",
[ "loop left Bol", "7.4", [ 7, 4, 0 ], 170, 40, "loop left bol",
"X780D907986EBA6C7" ],
[ "Bol loop right", "7.4", [ 7, 4, 0 ], 170, 39, "bol loop right",
[ "Bol loop right", "7.4", [ 7, 4, 0 ], 170, 40, "bol loop right",
"X780D907986EBA6C7" ],
[ "loop right Bol", "7.4", [ 7, 4, 0 ], 170, 39, "loop right bol",
[ "loop right Bol", "7.4", [ 7, 4, 0 ], 170, 40, "loop right bol",
"X780D907986EBA6C7" ],
[ "LC loop", "7.4", [ 7, 4, 0 ], 170, 39, "lc loop", "X780D907986EBA6C7" ],
[ "loop LC", "7.4", [ 7, 4, 0 ], 170, 39, "loop lc", "X780D907986EBA6C7" ],
[ "RC loop", "7.4", [ 7, 4, 0 ], 170, 39, "rc loop", "X780D907986EBA6C7" ],
[ "loop RC", "7.4", [ 7, 4, 0 ], 170, 39, "loop rc", "X780D907986EBA6C7" ],
[ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 39, "moufang loop",
[ "LC loop", "7.4", [ 7, 4, 0 ], 170, 40, "lc loop", "X780D907986EBA6C7" ],
[ "loop LC", "7.4", [ 7, 4, 0 ], 170, 40, "loop lc", "X780D907986EBA6C7" ],
[ "RC loop", "7.4", [ 7, 4, 0 ], 170, 40, "rc loop", "X780D907986EBA6C7" ],
[ "loop RC", "7.4", [ 7, 4, 0 ], 170, 40, "loop rc", "X780D907986EBA6C7" ],
[ "Moufang loop", "7.4", [ 7, 4, 0 ], 170, 40, "moufang loop",
"X780D907986EBA6C7" ],
[ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 39, "loop moufang",
[ "loop Moufang", "7.4", [ 7, 4, 0 ], 170, 40, "loop moufang",
"X780D907986EBA6C7" ],
[ "C loop", "7.4", [ 7, 4, 0 ], 170, 39, "c loop", "X780D907986EBA6C7" ],
[ "loop C", "7.4", [ 7, 4, 0 ], 170, 39, "loop c", "X780D907986EBA6C7" ],
[ "extra loop", "7.4", [ 7, 4, 0 ], 170, 39, "extra loop",
[ "C loop", "7.4", [ 7, 4, 0 ], 170, 40, "c loop", "X780D907986EBA6C7" ],
[ "loop C", "7.4", [ 7, 4, 0 ], 170, 40, "loop c", "X780D907986EBA6C7" ],
[ "extra loop", "7.4", [ 7, 4, 0 ], 170, 40, "extra loop",
"X780D907986EBA6C7" ],
[ "loop extra", "7.4", [ 7, 4, 0 ], 170, 39, "loop extra",
[ "loop extra", "7.4", [ 7, 4, 0 ], 170, 40, "loop extra",
"X780D907986EBA6C7" ],
[ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 39, "alternative loop",
[ "alternative loop", "7.4", [ 7, 4, 0 ], 170, 40, "alternative loop",
"X780D907986EBA6C7" ],
[ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 39, "loop alternative",
[ "loop alternative", "7.4", [ 7, 4, 0 ], 170, 40, "loop alternative",
"X780D907986EBA6C7" ],
[ "nuclear square loop", "7.4", [ 7, 4, 0 ], 170, 39, "nuclear square loop",
[ "nuclear square loop", "7.4", [ 7, 4, 0 ], 170, 40, "nuclear square loop",
"X780D907986EBA6C7" ],
[ "loop nuclear square", "7.4", [ 7, 4, 0 ], 170, 39, "loop nuclear square",
[ "loop nuclear square", "7.4", [ 7, 4, 0 ], 170, 40, "loop nuclear square",
"X780D907986EBA6C7" ],
[ "\033[2XIsExtraLoop\033[102X", "7.4-1", [ 7, 4, 1 ], 223, 40,
[ "\033[2XIsExtraLoop\033[102X", "7.4-1", [ 7, 4, 1 ], 223, 41,
"isextraloop", "X7988AFE27D06ACB5" ],
[ "\033[2XIsMoufangLoop\033[102X", "7.4-2", [ 7, 4, 2 ], 228, 40,
[ "\033[2XIsMoufangLoop\033[102X", "7.4-2", [ 7, 4, 2 ], 228, 41,
"ismoufangloop", "X7F1C151484C97E61" ],
[ "\033[2XIsCLoop\033[102X", "7.4-3", [ 7, 4, 3 ], 233, 40, "iscloop",
[ "\033[2XIsCLoop\033[102X", "7.4-3", [ 7, 4, 3 ], 233, 41, "iscloop",
"X866F04DC7AE54B7C" ],
[ "\033[2XIsLeftBolLoop\033[102X", "7.4-4", [ 7, 4, 4 ], 238, 40,
[ "\033[2XIsLeftBolLoop\033[102X", "7.4-4", [ 7, 4, 4 ], 238, 41,
"isleftbolloop", "X801DAAE8834A1A65" ],
[ "\033[2XIsRightBolLoop\033[102X", "7.4-5", [ 7, 4, 5 ], 243, 40,
[ "\033[2XIsRightBolLoop\033[102X", "7.4-5", [ 7, 4, 5 ], 243, 41,
"isrightbolloop", "X79279F9787E72566" ],
[ "\033[2XIsLCLoop\033[102X", "7.4-6", [ 7, 4, 6 ], 248, 40, "islcloop",
[ "\033[2XIsLCLoop\033[102X", "7.4-6", [ 7, 4, 6 ], 248, 41, "islcloop",
"X789E0A6979697C4C" ],
[ "\033[2XIsRCLoop\033[102X", "7.4-7", [ 7, 4, 7 ], 253, 40, "isrcloop",
[ "\033[2XIsRCLoop\033[102X", "7.4-7", [ 7, 4, 7 ], 253, 41, "isrcloop",
"X7B03CC577802F4AB" ],
[ "\033[2XIsLeftNuclearSquareLoop\033[102X", "7.4-8", [ 7, 4, 8 ], 258, 40,
[ "\033[2XIsLeftNuclearSquareLoop\033[102X", "7.4-8", [ 7, 4, 8 ], 258, 41,
"isleftnuclearsquareloop", "X819F285887B5EB9E" ],
[ "\033[2XIsMiddleNuclearSquareLoop\033[102X", "7.4-9", [ 7, 4, 9 ], 263,
40, "ismiddlenuclearsquareloop", "X8474F55681244A8A" ],
41, "ismiddlenuclearsquareloop", "X8474F55681244A8A" ],
[ "\033[2XIsRightNuclearSquareLoop\033[102X", "7.4-10", [ 7, 4, 10 ], 268,
40, "isrightnuclearsquareloop", "X807B3B21825E3076" ],
[ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 41,
41, "isrightnuclearsquareloop", "X807B3B21825E3076" ],
[ "\033[2XIsNuclearSquareLoop\033[102X", "7.4-11", [ 7, 4, 11 ], 273, 42,
"isnuclearsquareloop", "X796650088213229B" ],
[ "\033[2XIsFlexible\033[102X", "7.4-12", [ 7, 4, 12 ], 278, 41,
[ "\033[2XIsFlexible\033[102X", "7.4-12", [ 7, 4, 12 ], 278, 42,
"isflexible", "X7C32851A7AF1C45F" ],
[ "\033[2XIsLeftAlternative\033[102X", "7.4-13", [ 7, 4, 13 ], 283, 41,
[ "\033[2XIsLeftAlternative\033[102X", "7.4-13", [ 7, 4, 13 ], 283, 42,
"isleftalternative", "X7DF0196786B9CE08" ],
[ "\033[2XIsRightAlternative\033[102X", "7.4-14", [ 7, 4, 14 ], 288, 41,
[ "\033[2XIsRightAlternative\033[102X", "7.4-14", [ 7, 4, 14 ], 288, 42,
"isrightalternative", "X8416FAD87F148F5D" ],
[ "\033[2XIsAlternative\033[102X", "7.4-15", [ 7, 4, 15 ], 293, 41,
[ "\033[2XIsAlternative\033[102X", "7.4-15", [ 7, 4, 15 ], 293, 42,
"isalternative", "X8379356E82DB5DDA" ],
[ "power alternative loop left", "7.5", [ 7, 5, 0 ], 324, 42,
[ "power alternative loop left", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop left", "X83A501387E1AC371" ],
[ "loop left power alternative", "7.5", [ 7, 5, 0 ], 324, 42,
[ "loop left power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop left power alternative", "X83A501387E1AC371" ],
[ "power alternative loop right", "7.5", [ 7, 5, 0 ], 324, 42,
[ "power alternative loop right", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop right", "X83A501387E1AC371" ],
[ "loop right power alternative", "7.5", [ 7, 5, 0 ], 324, 42,
[ "loop right power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop right power alternative", "X83A501387E1AC371" ],
[ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 42,
[ "power alternative loop", "7.5", [ 7, 5, 0 ], 324, 43,
"power alternative loop", "X83A501387E1AC371" ],
[ "loop power alternative", "7.5", [ 7, 5, 0 ], 324, 42,
[ "loop power alternative", "7.5", [ 7, 5, 0 ], 324, 43,
"loop power alternative", "X83A501387E1AC371" ],
[ "\033[2XIsLeftPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42,
[ "\033[2XIsLeftPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"isleftpoweralternative", "X875C3DF681B3FAE2" ],
[ "\033[2XIsRightPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42,
[ "\033[2XIsRightPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"isrightpoweralternative", "X875C3DF681B3FAE2" ],
[ "\033[2XIsPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 42,
[ "\033[2XIsPowerAlternative\033[102X", "7.5-1", [ 7, 5, 1 ], 337, 43,
"ispoweralternative", "X875C3DF681B3FAE2" ],
[ "conjugacy closed loop left", "7.6", [ 7, 6, 0 ], 346, 42,
[ "conjugacy closed loop left", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop left", "X8176B2C47A4629CD" ],
[ "loop left conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42,
[ "loop left conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop left conjugacy closed", "X8176B2C47A4629CD" ],
[ "conjugacy closed loop right", "7.6", [ 7, 6, 0 ], 346, 42,
[ "conjugacy closed loop right", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop right", "X8176B2C47A4629CD" ],
[ "loop right conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42,
[ "loop right conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop right conjugacy closed", "X8176B2C47A4629CD" ],
[ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 42,
[ "conjugacy closed loop", "7.6", [ 7, 6, 0 ], 346, 43,
"conjugacy closed loop", "X8176B2C47A4629CD" ],
[ "loop conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 42,
[ "loop conjugacy closed", "7.6", [ 7, 6, 0 ], 346, 43,
"loop conjugacy closed", "X8176B2C47A4629CD" ],
[ "\033[2XIsLCCLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358, 42, "islccloop",
[ "\033[2XIsLCCLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358, 43, "islccloop",
"X784E08CD7B710AF4" ],
[ "\033[2XIsLeftConjugacyClosedLoop\033[102X", "7.6-1", [ 7, 6, 1 ], 358,
42, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ],
[ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 42, "isrccloop",
43, "isleftconjugacyclosedloop", "X784E08CD7B710AF4" ],
[ "\033[2XIsRCCLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364, 43, "isrccloop",
"X7B3016B47A1A8213" ],
[ "\033[2XIsRightConjugacyClosedLoop\033[102X", "7.6-2", [ 7, 6, 2 ], 364,
42, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ],
[ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 42, "isccloop",
43, "isrightconjugacyclosedloop", "X7B3016B47A1A8213" ],
[ "\033[2XIsCCLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 43, "isccloop",
"X878B614479DCB83F" ],
[ "\033[2XIsConjugacyClosedLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 42,
[ "\033[2XIsConjugacyClosedLoop\033[102X", "7.6-3", [ 7, 6, 3 ], 370, 43,
"isconjugacyclosedloop", "X878B614479DCB83F" ],
[ "\033[2XIsOsbornLoop\033[102X", "7.6-4", [ 7, 6, 4 ], 376, 42,
[ "\033[2XIsOsbornLoop\033[102X", "7.6-4", [ 7, 6, 4 ], 376, 43,
"isosbornloop", "X8655956878205FC1" ],
[ "Osborn loop", "7.6-4", [ 7, 6, 4 ], 376, 42, "osborn loop",
[ "Osborn loop", "7.6-4", [ 7, 6, 4 ], 376, 43, "osborn loop",
"X8655956878205FC1" ],
[ "loop Osborn", "7.6-4", [ 7, 6, 4 ], 376, 42, "loop osborn",
[ "loop Osborn", "7.6-4", [ 7, 6, 4 ], 376, 43, "loop osborn",
"X8655956878205FC1" ],
[ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop left", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop left", "X793B22EA8643C667" ],
[ "loop left automorphic", "7.7", [ 7, 7, 0 ], 384, 43,
[ "loop left automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop left automorphic", "X793B22EA8643C667" ],
[ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop middle", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop middle", "X793B22EA8643C667" ],
[ "loop middle automorphic", "7.7", [ 7, 7, 0 ], 384, 43,
[ "loop middle automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop middle automorphic", "X793B22EA8643C667" ],
[ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 43,
[ "automorphic loop right", "7.7", [ 7, 7, 0 ], 384, 44,
"automorphic loop right", "X793B22EA8643C667" ],
[ "loop right automorphic", "7.7", [ 7, 7, 0 ], 384, 43,
[ "loop right automorphic", "7.7", [ 7, 7, 0 ], 384, 44,
"loop right automorphic", "X793B22EA8643C667" ],
[ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 43, "automorphic loop",
[ "automorphic loop", "7.7", [ 7, 7, 0 ], 384, 44, "automorphic loop",
"X793B22EA8643C667" ],
[ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 43, "loop automorphic",
[ "loop automorphic", "7.7", [ 7, 7, 0 ], 384, 44, "loop automorphic",
"X793B22EA8643C667" ],
[ "\033[2XIsLeftAutomorphicLoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 43,
[ "\033[2XIsLeftAutomorphicLoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 44,
"isleftautomorphicloop", "X7F063914804659F1" ],
[ "\033[2XIsLeftALoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 43,
[ "\033[2XIsLeftALoop\033[102X", "7.7-1", [ 7, 7, 1 ], 425, 44,
"isleftaloop", "X7F063914804659F1" ],
[ "\033[2XIsMiddleAutomorphicLoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 43,
[ "\033[2XIsMiddleAutomorphicLoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 44,
"ismiddleautomorphicloop", "X7DFE830584A769E5" ],
[ "\033[2XIsMiddleALoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 43,
[ "\033[2XIsMiddleALoop\033[102X", "7.7-2", [ 7, 7, 2 ], 431, 44,
"ismiddlealoop", "X7DFE830584A769E5" ],
[ "\033[2XIsRightAutomorphicLoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 44,
[ "\033[2XIsRightAutomorphicLoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 45,
"isrightautomorphicloop", "X7EA9165A87F99E35" ],
[ "\033[2XIsRightALoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 44,
[ "\033[2XIsRightALoop\033[102X", "7.7-3", [ 7, 7, 3 ], 437, 45,
"isrightaloop", "X7EA9165A87F99E35" ],
[ "\033[2XIsAutomorphicLoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 44,
[ "\033[2XIsAutomorphicLoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 45,
"isautomorphicloop", "X7899603184CF13FD" ],
[ "\033[2XIsALoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 44, "isaloop",
[ "\033[2XIsALoop\033[102X", "7.7-4", [ 7, 7, 4 ], 443, 45, "isaloop",
"X7899603184CF13FD" ],
[ "\033[2XIsCodeLoop\033[102X", "7.8-1", [ 7, 8, 1 ], 454, 44,
[ "\033[2XIsCodeLoop\033[102X", "7.8-1", [ 7, 8, 1 ], 454, 45,
"iscodeloop", "X790FA1188087D5C1" ],
[ "code loop", "7.8-1", [ 7, 8, 1 ], 454, 44, "code loop",
[ "code loop", "7.8-1", [ 7, 8, 1 ], 454, 45, "code loop",
"X790FA1188087D5C1" ],
[ "loop code", "7.8-1", [ 7, 8, 1 ], 454, 44, "loop code",
[ "loop code", "7.8-1", [ 7, 8, 1 ], 454, 45, "loop code",
"X790FA1188087D5C1" ],
[ "\033[2XIsSteinerLoop\033[102X", "7.8-2", [ 7, 8, 2 ], 462, 44,
[ "\033[2XIsSteinerLoop\033[102X", "7.8-2", [ 7, 8, 2 ], 462, 45,
"issteinerloop", "X793600C9801F4F62" ],
[ "Steiner loop", "7.8-2", [ 7, 8, 2 ], 462, 44, "steiner loop",
[ "Steiner loop", "7.8-2", [ 7, 8, 2 ], 462, 45, "steiner loop",
"X793600C9801F4F62" ],
[ "loop Steiner", "7.8-2", [ 7, 8, 2 ], 462, 44, "loop steiner",
[ "loop Steiner", "7.8-2", [ 7, 8, 2 ], 462, 45, "loop steiner",
"X793600C9801F4F62" ],
[ "\033[2XIsLeftBruckLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 44,
[ "\033[2XIsLeftBruckLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 45,
"isleftbruckloop", "X85F1BD4280E44F5B" ],
[ "\033[2XIsLeftKLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 44,
[ "\033[2XIsLeftKLoop\033[102X", "7.8-3", [ 7, 8, 3 ], 470, 45,
"isleftkloop", "X85F1BD4280E44F5B" ],
[ "Bruck loop left", "7.8-3", [ 7, 8, 3 ], 470, 44, "bruck loop left",
[ "Bruck loop left", "7.8-3", [ 7, 8, 3 ], 470, 45, "bruck loop left",
"X85F1BD4280E44F5B" ],
[ "loop left Bruck", "7.8-3", [ 7, 8, 3 ], 470, 44, "loop left bruck",
[ "loop left Bruck", "7.8-3", [ 7, 8, 3 ], 470, 45, "loop left bruck",
"X85F1BD4280E44F5B" ],
[ "K loop left", "7.8-3", [ 7, 8, 3 ], 470, 44, "k loop left",
[ "K loop left", "7.8-3", [ 7, 8, 3 ], 470, 45, "k loop left",
"X85F1BD4280E44F5B" ],
[ "loop left K", "7.8-3", [ 7, 8, 3 ], 470, 44, "loop left k",
[ "loop left K", "7.8-3", [ 7, 8, 3 ], 470, 45, "loop left k",
"X85F1BD4280E44F5B" ],
[ "\033[2XIsRightBruckLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 44,
[ "\033[2XIsRightBruckLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 45,
"isrightbruckloop", "X857B373E7B4E0519" ],
[ "\033[2XIsRightKLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 44,
[ "\033[2XIsRightKLoop\033[102X", "7.8-4", [ 7, 8, 4 ], 480, 45,
"isrightkloop", "X857B373E7B4E0519" ],
[ "Bruck loop right", "7.8-4", [ 7, 8, 4 ], 480, 44, "bruck loop right",
[ "Bruck loop right", "7.8-4", [ 7, 8, 4 ], 480, 45, "bruck loop right",
"X857B373E7B4E0519" ],
[ "loop right Bruck", "7.8-4", [ 7, 8, 4 ], 480, 44, "loop right bruck",
[ "loop right Bruck", "7.8-4", [ 7, 8, 4 ], 480, 45, "loop right bruck",
"X857B373E7B4E0519" ],
[ "K loop right", "7.8-4", [ 7, 8, 4 ], 480, 44, "k loop right",
[ "K loop right", "7.8-4", [ 7, 8, 4 ], 480, 45, "k loop right",
"X857B373E7B4E0519" ],
[ "loop right K", "7.8-4", [ 7, 8, 4 ], 480, 44, "loop right k",
[ "loop right K", "7.8-4", [ 7, 8, 4 ], 480, 45, "loop right k",
"X857B373E7B4E0519" ],
[ "\033[2XAssociatedLeftBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 45,
[ "\033[2XAssociatedLeftBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 46,
"associatedleftbruckloop", "X8664CA927DD73DBE" ],
[ "\033[2XAssociatedRightBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 45,
[ "\033[2XAssociatedRightBruckLoop\033[102X", "8.1-1", [ 8, 1, 1 ], 10, 46,
"associatedrightbruckloop", "X8664CA927DD73DBE" ],
[ "loop left Bol", "8.1-1", [ 8, 1, 1 ], 10, 45, "loop left bol",
[ "loop left Bol", "8.1-1", [ 8, 1, 1 ], 10, 46, "loop left bol",
"X8664CA927DD73DBE" ],
[ "Bol loop left", "8.1-1", [ 8, 1, 1 ], 10, 45, "bol loop left",
[ "Bol loop left", "8.1-1", [ 8, 1, 1 ], 10, 46, "bol loop left",
"X8664CA927DD73DBE" ],
[ "Bruck loop associated left", "8.1-1", [ 8, 1, 1 ], 10, 45,
[ "Bruck loop associated left", "8.1-1", [ 8, 1, 1 ], 10, 46,
"bruck loop associated left", "X8664CA927DD73DBE" ],
[ "loop associated left Bruck", "8.1-1", [ 8, 1, 1 ], 10, 45,
[ "loop associated left Bruck", "8.1-1", [ 8, 1, 1 ], 10, 46,
"loop associated left bruck", "X8664CA927DD73DBE" ],
[ "\033[2XIsExactGroupFactorization\033[102X", "8.1-2", [ 8, 1, 2 ], 26,
45, "isexactgroupfactorization", "X82FC16F386CE11F1" ],
[ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 45,
46, "isexactgroupfactorization", "X82FC16F386CE11F1" ],
[ "exact group factorization", "8.1-2", [ 8, 1, 2 ], 26, 46,
"exact group factorization", "X82FC16F386CE11F1" ],
[ "\033[2XRightBolLoopByExactGroupFactorization\033[102X", "8.1-3",
[ 8, 1, 3 ], 35, 45, "rightbolloopbyexactgroupfactorization",
[ 8, 1, 3 ], 35, 46, "rightbolloopbyexactgroupfactorization",
"X7DCA64807F899127" ],
[ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 46,
[ "modification Moufang", "8.2", [ 8, 2, 0 ], 47, 47,
"modification moufang", "X819F82737C2A860D" ],
[ "\033[2XLoopByCyclicModification\033[102X", "8.2-1", [ 8, 2, 1 ], 57, 46,
[ "\033[2XLoopByCyclicModification\033[102X", "8.2-1", [ 8, 2, 1 ], 57, 47,
"loopbycyclicmodification", "X7B3165C083709831" ],
[ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 46,
[ "modification cyclic", "8.2-1", [ 8, 2, 1 ], 57, 47,
"modification cyclic", "X7B3165C083709831" ],
[ "\033[2XLoopByDihedralModification\033[102X", "8.2-2", [ 8, 2, 2 ], 70,
46, "loopbydihedralmodification", "X7D7717C587BC2D1E" ],
[ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 46,
47, "loopbydihedralmodification", "X7D7717C587BC2D1E" ],
[ "modification dihedral", "8.2-2", [ 8, 2, 2 ], 70, 47,
"modification dihedral", "X7D7717C587BC2D1E" ],
[ "\033[2XLoopMG2\033[102X", "8.2-3", [ 8, 2, 3 ], 86, 46, "loopmg2",
[ "\033[2XLoopMG2\033[102X", "8.2-3", [ 8, 2, 3 ], 86, 47, "loopmg2",
"X7CC6CDB786E9BBA0" ],
[ "Chein loop", "8.2-3", [ 8, 2, 3 ], 86, 46, "chein loop",
[ "Chein loop", "8.2-3", [ 8, 2, 3 ], 86, 47, "chein loop",
"X7CC6CDB786E9BBA0" ],
[ "loop Chein", "8.2-3", [ 8, 2, 3 ], 86, 46, "loop chein",
[ "loop Chein", "8.2-3", [ 8, 2, 3 ], 86, 47, "loop chein",
"X7CC6CDB786E9BBA0" ],
[ "group with triality", "8.3", [ 8, 3, 0 ], 98, 46, "group with triality",
[ "group with triality", "8.3", [ 8, 3, 0 ], 98, 47, "group with triality",
"X83E73A767D79FAFD" ],
[ "\033[2XTrialityPermGroup\033[102X", "8.3-1", [ 8, 3, 1 ], 113, 47,
[ "\033[2XTrialityPermGroup\033[102X", "8.3-1", [ 8, 3, 1 ], 113, 48,
"trialitypermgroup", "X7DB4DE647F6F56F0" ],
[ "\033[2XTrialityPcGroup\033[102X", "8.3-2", [ 8, 3, 2 ], 120, 47,
[ "\033[2XTrialityPcGroup\033[102X", "8.3-2", [ 8, 3, 2 ], 120, 48,
"trialitypcgroup", "X82CC977085DFDFE8" ],
[ "\033[2XAllLoopTablesInGroup\033[102X", "8.4-1", [ 8, 4, 1 ], 146, 47,
[ "\033[2XAllLoopTablesInGroup\033[102X", "8.4-1", [ 8, 4, 1 ], 146, 48,
"alllooptablesingroup", "X804F40087DD1225D" ],
[ "\033[2XAllProperLoopTablesInGroup\033[102X", "8.4-2", [ 8, 4, 2 ], 152,
47, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ],
[ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 47,
48, "allproperlooptablesingroup", "X7854C8E382DC8E8B" ],
[ "\033[2XOneLoopTableInGroup\033[102X", "8.4-3", [ 8, 4, 3 ], 158, 48,
"onelooptableingroup", "X7BFFC66A824BA6AA" ],
[ "\033[2XOneProperLoopTableInGroup\033[102X", "8.4-4", [ 8, 4, 4 ], 164,
48, "oneproperlooptableingroup", "X84C5A76585B335FF" ],
[ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 48,
49, "oneproperlooptableingroup", "X84C5A76585B335FF" ],
[ "\033[2XAllLoopsWithMltGroup\033[102X", "8.4-5", [ 8, 4, 5 ], 170, 49,
"allloopswithmltgroup", "X7E5F1C2879358EEF" ],
[ "\033[2XOneLoopWithMltGroup\033[102X", "8.4-6", [ 8, 4, 6 ], 176, 48,
[ "\033[2XOneLoopWithMltGroup\033[102X", "8.4-6", [ 8, 4, 6 ], 176, 49,
"oneloopwithmltgroup", "X8266DE05824226E6" ],
[ "\033[2XLibraryLoop\033[102X", "9.1-1", [ 9, 1, 1 ], 31, 49,
[ "\033[2XLibraryLoop\033[102X", "9.1-1", [ 9, 1, 1 ], 31, 50,
"libraryloop", "X849865D6786EEF9B" ],
[ "\033[2XMyLibraryLoop\033[102X", "9.1-2", [ 9, 1, 2 ], 36, 49,
[ "\033[2XMyLibraryLoop\033[102X", "9.1-2", [ 9, 1, 2 ], 36, 50,
"mylibraryloop", "X78C4B8757902D49F" ],
[ "\033[2XDisplayLibraryInfo\033[102X", "9.1-3", [ 9, 1, 3 ], 46, 50,
[ "\033[2XDisplayLibraryInfo\033[102X", "9.1-3", [ 9, 1, 3 ], 46, 51,
"displaylibraryinfo", "X7A64372E81E713B4" ],
[ "\033[2XLeftBolLoop\033[102X", "9.2-1", [ 9, 2, 1 ], 67, 50,
[ "\033[2XLeftBolLoop\033[102X", "9.2-1", [ 9, 2, 1 ], 67, 51,
"leftbolloop", "X7EE99F647C537994" ],
[ "\033[2XRightBolLoop\033[102X", "9.2-2", [ 9, 2, 2 ], 72, 50,
[ "\033[2XRightBolLoop\033[102X", "9.2-2", [ 9, 2, 2 ], 72, 51,
"rightbolloop", "X8774304282654C58" ],
[ "\033[2XMoufangLoop\033[102X", "9.3-1", [ 9, 3, 1 ], 86, 50,
[ "\033[2XLeftBruckLoop\033[102X", "9.3-1", [ 9, 3, 1 ], 92, 51,
"leftbruckloop", "X8290B01780F0FCD3" ],
[ "\033[2XRightBruckLoop\033[102X", "9.3-2", [ 9, 3, 2 ], 97, 51,
"rightbruckloop", "X798DD7CF871F648F" ],
[ "\033[2XMoufangLoop\033[102X", "9.4-1", [ 9, 4, 1 ], 108, 52,
"moufangloop", "X81E82098822543EE" ],
[ "octonion loop", "9.3-1", [ 9, 3, 1 ], 86, 50, "octonion loop",
[ "octonion loop", "9.4-1", [ 9, 4, 1 ], 108, 52, "octonion loop",
"X81E82098822543EE" ],
[ "loop octonion", "9.3-1", [ 9, 3, 1 ], 86, 50, "loop octonion",
[ "loop octonion", "9.4-1", [ 9, 4, 1 ], 108, 52, "loop octonion",
"X81E82098822543EE" ],
[ "\033[2XCodeLoop\033[102X", "9.4-1", [ 9, 4, 1 ], 117, 51, "codeloop",
[ "\033[2XCodeLoop\033[102X", "9.5-1", [ 9, 5, 1 ], 139, 52, "codeloop",
"X7DB4D3B27BB4D7EE" ],
[ "\033[2XSteinerLoop\033[102X", "9.5-1", [ 9, 5, 1 ], 144, 51,
[ "\033[2XSteinerLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 166, 53,
"steinerloop", "X87C235457E859AF4" ],
[ "\033[2XRCCLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 173, 52, "rccloop",
[ "\033[2XRCCLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 195, 53, "rccloop",
"X806B2DE67990E42F" ],
[ "\033[2XRightConjugacyClosedLoop\033[102X", "9.6-1", [ 9, 6, 1 ], 173,
52, "rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[ "\033[2XLCCLoop\033[102X", "9.6-2", [ 9, 6, 2 ], 180, 52, "lccloop",
[ "\033[2XRightConjugacyClosedLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 195,
53, "rightconjugacyclosedloop", "X806B2DE67990E42F" ],
[ "\033[2XLCCLoop\033[102X", "9.7-2", [ 9, 7, 2 ], 202, 53, "lccloop",
"X80AB8B107D55FB19" ],
[ "\033[2XLeftConjugacyClosedLoop\033[102X", "9.6-2", [ 9, 6, 2 ], 180, 52,
[ "\033[2XLeftConjugacyClosedLoop\033[102X", "9.7-2", [ 9, 7, 2 ], 202, 53,
"leftconjugacyclosedloop", "X80AB8B107D55FB19" ],
[ "\033[2XCCLoop\033[102X", "9.6-3", [ 9, 6, 3 ], 217, 52, "ccloop",
[ "\033[2XCCLoop\033[102X", "9.7-3", [ 9, 7, 3 ], 241, 54, "ccloop",
"X798BC601843E8916" ],
[ "\033[2XConjugacyClosedLoop\033[102X", "9.6-3", [ 9, 6, 3 ], 217, 52,
[ "\033[2XConjugacyClosedLoop\033[102X", "9.7-3", [ 9, 7, 3 ], 241, 54,
"conjugacyclosedloop", "X798BC601843E8916" ],
[ "\033[2XSmallLoop\033[102X", "9.7-1", [ 9, 7, 1 ], 230, 53, "smallloop",
[ "\033[2XSmallLoop\033[102X", "9.8-1", [ 9, 8, 1 ], 254, 54, "smallloop",
"X7C6EE23E84CD87D3" ],
[ "Paige loop", "9.8", [ 9, 8, 0 ], 235, 53, "paige loop",
[ "Paige loop", "9.9", [ 9, 9, 0 ], 259, 54, "paige loop",
"X8135C8FD8714C606" ],
[ "loop Paige", "9.8", [ 9, 8, 0 ], 235, 53, "loop paige",
[ "loop Paige", "9.9", [ 9, 9, 0 ], 259, 54, "loop paige",
"X8135C8FD8714C606" ],
[ "\033[2XPaigeLoop\033[102X", "9.8-1", [ 9, 8, 1 ], 244, 53, "paigeloop",
[ "\033[2XPaigeLoop\033[102X", "9.9-1", [ 9, 9, 1 ], 268, 54, "paigeloop",
"X7FCF4D6B7AD66D74" ],
[ "\033[2XNilpotentLoop\033[102X", "9.9-1", [ 9, 9, 1 ], 261, 53,
[ "\033[2XNilpotentLoop\033[102X", "9.10-1", [ 9, 10, 1 ], 285, 54,
"nilpotentloop", "X7A9C960D86E2AD28" ],
[ "\033[2XAutomorphicLoop\033[102X", "9.10-1", [ 9, 10, 1 ], 278, 53,
[ "\033[2XAutomorphicLoop\033[102X", "9.11-1", [ 9, 11, 1 ], 304, 55,
"automorphicloop", "X784FFA9E7FDA9F43" ],
[ "sedenion loop", "9.11", [ 9, 11, 0 ], 283, 54, "sedenion loop",
[ "sedenion loop", "9.12", [ 9, 12, 0 ], 309, 55, "sedenion loop",
"X843BD73F788049F7" ],
[ "loop sedenion", "9.11", [ 9, 11, 0 ], 283, 54, "loop sedenion",
[ "loop sedenion", "9.12", [ 9, 12, 0 ], 309, 55, "loop sedenion",
"X843BD73F788049F7" ],
[ "\033[2XInterestingLoop\033[102X", "9.11-1", [ 9, 11, 1 ], 293, 54,
[ "\033[2XInterestingLoop\033[102X", "9.12-1", [ 9, 12, 1 ], 319, 55,
"interestingloop", "X87F24AD3811910D3" ],
[ "\033[2XItpSmallLoop\033[102X", "9.12-1", [ 9, 12, 1 ], 306, 54,
[ "\033[2XItpSmallLoop\033[102X", "9.13-1", [ 9, 13, 1 ], 332, 55,
"itpsmallloop", "X850C4C01817A098D" ] ]
);

6
etc/gapdoc.txt
View File

@ -22,7 +22,7 @@ chooser.html
When files are ready, run the following in GAP:
# path to files, change as needed
path := Directory("c:/cygwin64/opt/gap4r7/pkg/loops/doc");;
path := Directory("c:/cygwin64/opt/gap4r8/pkg/loops/doc");;
main := "loops.xml";;
files := [];;
bookname := "loops";;
@ -52,7 +52,7 @@ GAPDoc2HTMLPrintHTMLFiles(h, path);
# h := GAPDoc2HTML(r, path );;
# GAPDoc2HTMLPrintHTMLFiles(h, path);
# now produce .ps, .dvi from .tex,
# and copy loops.* as manual.* for extensions pdf, ps, dvi
# now produce .ps from .tex
# and copy loops.* as manual.* for extensions pdf, ps
# delete auxiliary files

4
gap/banner.g
View File

@ -2,7 +2,7 @@
##
#A banner.g loops G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: banner.g, v 3.3.0 2016/09/21 gap Exp $
#H @(#)$Id: banner.g, v 3.4.0 2017/10/27 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -12,7 +12,7 @@ if not QUIET and BANNER then
Print(
" ______________________________________________________\n",
" LOOPS: Computing with quasigroups and loops in GAP \n",
" version 3.3.0 \n",
" version 3.4.0 \n",
" Gabor P. Nagy & Petr Vojtechovsky \n",
" nagyg@math.u-szeged.hu petr@math.du.edu \n",
" ------------------------------------------------------\n",

12
gap/classes.gi
View File

@ -2,7 +2,7 @@
##
#W classes.gi Testing properties/varieties [loops]
##
#H @(#)$Id: classes.gi, v 3.3.0 2016/10/26 gap Exp $
#H @(#)$Id: classes.gi, v 3.4.0 2017/10/26 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -887,16 +887,14 @@ end);
InstallMethod( IsALoop, "for loop",
[ IsLoop ],
function( Q )
return IsLeftALoop(Q) and IsRightALoop(Q) and IsMiddleALoop(Q);
return IsRightALoop(Q) and IsMiddleALoop(Q);
# Theorem: rigth A-loop + middle A-loop implies left A-loop
end);
# implies
InstallTrueMethod( IsLeftALoop, IsALoop );
InstallTrueMethod( IsRightALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsALoop );
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsALoop, IsLeftALoop and IsCommutative );
InstallTrueMethod( IsALoop, IsRightALoop and IsCommutative );
InstallTrueMethod( IsLeftALoop, IsRightALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsRightALoop, IsLeftALoop and HasAntiautomorphicInverseProperty );
InstallTrueMethod( IsFlexible, IsMiddleALoop );
@ -909,8 +907,12 @@ InstallTrueMethod( IsMoufangLoop, IsALoop and HasRightInverseProperty );
InstallTrueMethod( IsMoufangLoop, IsALoop and HasWeakInverseProperty );
# is implied by
InstallTrueMethod( IsMiddleALoop, IsCommutative );
InstallTrueMethod( IsLeftALoop, IsLeftBruckLoop );
InstallTrueMethod( IsLeftALoop, IsLCCLoop );
InstallTrueMethod( IsRightALoop, IsRightBruckLoop );
InstallTrueMethod( IsRightALoop, IsRCCLoop );
InstallTrueMethod( IsALoop, IsCommutative and IsMoufangLoop );
InstallTrueMethod( IsALoop, IsLeftALoop and IsMiddleALoop );
InstallTrueMethod( IsALoop, IsRightALoop and IsMiddleALoop );
InstallTrueMethod( IsALoop, IsAssociative );

5
gap/examples.gd
View File

@ -2,7 +2,7 @@
##
#W examples.gd Examples [loops]
##
#H @(#)$Id: examples.gd, v 3.1.0 2015/09/23 gap Exp $
#H @(#)$Id: examples.gd, v 3.4.0 2015/09/23 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -36,6 +36,8 @@ DeclareGlobalFunction( "SmallLoop" );
DeclareGlobalFunction( "InterestingLoop" );
DeclareGlobalFunction( "NilpotentLoop" );
DeclareGlobalFunction( "AutomorphicLoop" );
DeclareGlobalFunction( "LeftBruckLoop" );
DeclareGlobalFunction( "RightBruckLoop" );
# up to isotopism
@ -52,3 +54,4 @@ DeclareGlobalFunction( "LOOPS_ActivateRCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateCCLoop" );
DeclareGlobalFunction( "LOOPS_ActivateNilpotentLoop" );
DeclareGlobalFunction( "LOOPS_ActivateAutomorphicLoop" );
DeclareGlobalFunction( "LOOPS_ActivateRightBruckLoop" );

213
gap/examples.gi
View File

@ -2,7 +2,7 @@
##
#W examples.gi Examples [loops]
##
#H @(#)$Id: examples.gi, v 3.3.0 2016/10/19 gap Exp $
#H @(#)$Id: examples.gi, v 3.4.0 2017/10/23 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -32,6 +32,7 @@ ReadPackage("loops", "data/small.tbl"); # small loops
ReadPackage("loops", "data/interesting.tbl"); # interesting loops
ReadPackage("loops", "data/nilpotent.tbl"); # nilpotent loops
ReadPackage("loops", "data/automorphic.tbl"); # automorphic loops
ReadPackage("loops", "data/rightbruck.tbl"); # right Bruck loops
# up to isotopism
ReadPackage("loops", "data/itp_small.tbl"); # small loops up to isotopism
@ -60,6 +61,7 @@ function( name )
elif name = "interesting" then return LOOPS_interesting_data;
elif name = "nilpotent" then return LOOPS_nilpotent_data;
elif name = "automorphic" then return LOOPS_automorphic_data;
elif name = "right Bruck" then return LOOPS_right_bruck_data;
#up to isotopism
elif name = "itp small" then return LOOPS_itp_small_data;
fi;
@ -74,11 +76,8 @@ end);
InstallGlobalFunction( DisplayLibraryInfo, function( name )
local s, lib, k;
# up to isomorphism
if name = "left Bol" then
s := "The library contains all nonassociative left Bol loops of order less than 17\nand all nonassociative left Bol loops of order p*q, where p>q>2 are primes.";
elif name = "right Bol" then
s := "The library contains all nonassociative right Bol loops of order less than 17\nand all nonassociative left Bol loops of order p*q, where p>q>2 are primes.";
name := "left Bol"; # using dual data
if name = "left Bol" or name = "right Bol" then
s := Concatenation( "The library contains all nonassociative ", name, " loops of order less than 17\nand all nonassociative ", name, " loops of order p*q, where p>q>2 are primes." );
elif name = "Moufang" then
s := "The library contains all nonassociative Moufang loops \nof order less than 65, and all nonassociative Moufang \nloops of order 81 and 243.";
elif name = "Paige" then
@ -88,12 +87,9 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
elif name = "Steiner" then
s := "The library contains all nonassociative Steiner loops \nof order less or equal to 16. It also contains the \nassociative Steiner loops of order 4 and 8.";
elif name = "CC" then
s := "The library contains all nonassociative CC loops of order less than 28 \nand all nonassociative CC loops of order p^2 and 2*p for any odd prime p.";
elif name = "RCC" then
s := "The library contains all nonassociative RCC loops of order less than 28.";
elif name = "LCC" then
s := "The library contains all nonassociative LCC loops of order less than 28.";
name := "RCC"; # using dual data
s := "The library contains all CC loops of order\n2<=2^k<=64, 3<=3^k<=81, 5<=5^k<=125, 7<=7^k<=343,\nall nonassociative CC loops of order less than 28,\nand all nonassociative CC loops of order p^2 and 2*p for any odd prime p.";
elif name = "RCC" or name = "LCC" then
s := Concatenation( "The library contains all nonassociative ", name, " loops of order less than 28." );
elif name = "small" then
s := "The library contains all nonassociative loops of order less than 7.";
elif name = "interesting" then
@ -103,23 +99,27 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
elif name = "automorphic" then
s := "The library contains:\n";
s := Concatenation(s," - all nonassociative automorphic loops of order less than 16,\n");
s := Concatenation(s," - all commutative automorphic loops of order 3, 9, 27, 81,\n");
s := Concatenation(s," - all commutative automorphic loops of order 243 that are central\n");
s := Concatenation(s," extensions of Z_3 by F, where F is not the elem. ab. 3-group.\n");
s := Concatenation(s,"Note: Abelian groups are included among the commutative loops.");
s := Concatenation(s," - all commutative automorphic loops of order 3, 9, 27, 81.");
elif name = "left Bruck" or name = "right Bruck" then
s := Concatenation( "The library contains all ", name, " loops of orders 3, 9, 27 and 81." );
# up to isotopism
elif name = "itp small" then
s := "The library contains all nonassociative loops of order less than 7 up to isotopism.";
else
Info( InfoWarning, 1, Concatenation(
"The admissible names for loop libraries are: \n",
"[ \"left Bol\", \"right Bol\", \"Moufang\", \"Paige\", \"code\", \"Steiner\", \"CC\", \"RCC\", \"LCC\", \"small\", \"itp small\", \"interesting\", \"nilpotent\", \"automorphic\" ]."
"\"automorphic\", \"CC\", \"code\", \"interesting\", \"itp small\", \"LCC\", \"left Bol\", \"left Bruck\", \"Moufang\", \"nilpotent\", \"Paige\", \"right Bol\", \"right Bruck\", \"RCC\", \"small\", \"Steiner\"."
) );
return fail;
fi;
s := Concatenation( s, "\n------\nExtent of the library:" );
# renaming for data access
if name = "right Bol" then name := "left Bol"; fi;
if name = "LCC" then name := "RCC"; fi;
if name = "left Bruck" then name := "right Bruck"; fi;
lib := LOOPS_LibraryByName( name );
for k in [1..Length( lib[ 1 ] ) ] do
if lib[ 2 ][ k ] = 1 then
@ -128,12 +128,12 @@ InstallGlobalFunction( DisplayLibraryInfo, function( name )
s := Concatenation( s, "\n ", String( lib[ 2 ][ k ] ), " loops of order ", String( lib[ 1 ][ k ] ) );
fi;
od;
if name = "left Bol" or name = "right Bol" then
if name = "left Bol" then
s := Concatenation( s, "\n (p-q)/2 loops of order p*q for primes p>q>2 such that q divides p-1");
s := Concatenation( s, "\n (p-q+2)/2 loops of order p*q for primes p>q>2 such that q divides p+1" );
fi;
if name = "CC" then
s := Concatenation( s, "\n 3 loops of order p^2 for every odd prime p,\n 1 loop of order 2*p for every odd prime p" );
s := Concatenation( s, "\n 3 loops of order p^2 for every prime p>7,\n 1 loop of order 2*p for every odd prime p" );
fi;
s := Concatenation( s, "\n" );
Print( s );
@ -436,7 +436,48 @@ end);
InstallGlobalFunction( LOOPS_ActivateCCLoop,
function( n, pos_n, m, case )
local T, x, y, k, a, b, p;
local powers, p, i, k, F, basis, coords, coc, T, a, b, x, y;
powers := [ ,[4,8,16,32,64],[9,27,81],,[25,125],,[49,343]];
if n in Union( powers ) then # use cocycles
# determine p and position of n in database
p := Filtered([2,3,5,7], x -> n in powers[x])[1];
pos_n := Position( powers[p], n );
if not IsBound( LOOPS_cc_cocycles[p] ) then
# data not read yet, activate once
ReadPackage( "loops", Concatenation( "data/cc/cc_cocycles_", String(p), ".tbl" ) );
# decode cocycles and separate coordinates from a long string
for i in [1..Length(powers[p])] do
LOOPS_cc_cocycles[ p ][ i ] := List( LOOPS_cc_cocycles[ p ][ i ],
c -> LOOPS_DecodeCocycle( [ p^i, c[1], c[2] ], [0..p-1] )
);
LOOPS_cc_coordinates[ p ][ i ] := List( LOOPS_cc_coordinates[ p ][ i ],
c -> SplitString( c, " " )
);
od;
fi;
# data is now read
# determine position of loop in the database
k := 1;
while m > Length( LOOPS_cc_coordinates[ p ][ pos_n ][ k ] ) do
m := m - Length( LOOPS_cc_coordinates[ p ][ pos_n ][ k ] );
k := k + 1;
od;
# factor loop
F := CCLoop( n/p, LOOPS_cc_used_factors[ p ][ pos_n ][ k ] );
# basis
basis := List( LOOPS_cc_bases[ p ][ pos_n ][ k ],
i -> LOOPS_cc_cocycles[ p ][ pos_n ][ i ]
);
# coordinates
coords := LOOPS_cc_coordinates[ p ][ pos_n ][ k ][ m ];
coords := LOOPS_ConvertBase( coords, 91, p, Length( basis ) );
coords := List( coords, LOOPS_CharToDigit );
# cocycle
coc := (coords*basis) mod p;
coc := List( coc, i -> i+1 );
# return extension of Z_p by F using cocycle and trivial action
return LoopByExtension( CCLoop(p,1), F, List([1..n/p], i -> () ), coc );
fi;
if case=false then # use library of RCC loops, must recalculate pos_n
return LOOPS_ActivateRCCLoop( n, Position(LOOPS_rcc_data[ 1 ], n), LOOPS_cc_data[ 3 ][ pos_n ][ m ] );
@ -543,39 +584,52 @@ end);
InstallGlobalFunction( LOOPS_ActivateAutomorphicLoop,
function( n, m )
local i, pos_n, factor_id, F, dim, coords, basis, coc;
if IsEmpty( LOOPS_automorphic_cocycles ) then # only read on demand
ReadPackage( "loops", "data/automorphic/automorphic_cocycles.tbl");
# decode cocycles
for i in [1..3] do
LOOPS_automorphic_cocycles[ i ] := List( LOOPS_automorphic_cocycles[ i ],
c -> LOOPS_DecodeCocycle( [ 3^(i+2), true, c ], [0,1,2] )
);
od;
# separate coordinates (from a long string )
for i in [1..3] do
LOOPS_automorphic_coordinates[ i ] := SplitString( LOOPS_automorphic_coordinates[ i ], " " );
od;
fi;
# returns the associated Gamma loop (which here always happens to be automorphic)
# improve later
local P, L, s, Ls, ct, i, j, pos, f;
P := LeftBruckLoop( n, m );
L := LeftMultiplicationGroup( P );;
s := List(Elements(L), x -> x^2 );;
Ls := List([1..n], i -> LeftTranslation( P, Elements(P)[i] ) );;
ct := List([1..n],i->0*[1..n]);;
for i in [1..n] do for j in [1..n] do
pos := Position( s, Ls[i]*Ls[j]*Ls[i]^(-1)*Ls[j]^(-1) );
f := Elements(L)[pos];
ct[i][j] := 1^(f*Ls[j]*Ls[i]);
od; od;
return LoopByCayleyTable(ct);
end);
#############################################################################
##
#F LOOPS_ActivateRightBruckLoop( n, m )
##
## Activates a right Bruck loop from the library.
InstallGlobalFunction( LOOPS_ActivateRightBruckLoop,
function( n, m )
local pos_n, factor_id, F, basis, coords, coc;
# factor loop
pos_n := Position( [27,81,243], n );
factor_id := LOOPS_CharToDigit( LOOPS_automorphic_coordinates[ pos_n ][ m ][ 1 ] );
F := AutomorphicLoop( n/3, factor_id );
pos_n := Position( [27,81], n );
factor_id := LOOPS_CharToDigit( LOOPS_right_bruck_coordinates[ pos_n ][ m ][ 1 ] );
F := RightBruckLoop( n/3, factor_id );
# basis (only decode cocycles at first usage)
if IsString( LOOPS_right_bruck_cocycles[ pos_n ][ 1 ][ 3 ] ) then # not converted yet
LOOPS_right_bruck_cocycles[ pos_n ] := List( LOOPS_right_bruck_cocycles[ pos_n ],
coc -> LOOPS_DecodeCocycle( coc, [0,1,2] )
);
fi;
basis := LOOPS_right_bruck_cocycles[ pos_n ];
# coordinates determining the cocycle
dim := Length( LOOPS_automorphic_bases[ pos_n ][ factor_id ] );
coords := LOOPS_automorphic_coordinates[ pos_n ][ m ];
coords := LOOPS_right_bruck_coordinates[ pos_n ][ m ];
coords := coords{[2..Length(coords)]}; # remove the character that determines factor id
coords := LOOPS_ConvertBase( coords, 91, 3, dim );
coords := LOOPS_ConvertBase( coords, 91, 3, Length( basis ) );
coords := List( coords, LOOPS_CharToDigit );
# basis
basis := List( LOOPS_automorphic_bases[ pos_n ][ factor_id ],
i -> LOOPS_automorphic_cocycles[ pos_n ][ i ]
);
# calculate cocycle
coc := (coords*basis) mod 3;
coc := List( coc, i -> i+1 );
coc := coc + 1;
# return extension of Z_3 by F using cocycle and trivial action
return LoopByExtension( AutomorphicLoop(3,1), F, List([1..n/3], i -> () ), coc );
return LoopByExtension( RightBruckLoop(3,1), F, List([1..n/3], i -> () ), coc );
end);
#############################################################################
@ -593,13 +647,7 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
local lib, implemented_orders, NOL, loop, pos_n, p, q, divs, PG, m_inv, root, half, case, g, h;
# selecting data library
if name = "right Bol" then # using dual data
lib := LOOPS_LibraryByName( "left Bol" );
elif name = "LCC" then # using dual data
lib := LOOPS_LibraryByName( "RCC" );
else
lib := LOOPS_LibraryByName( name );
fi;
lib := LOOPS_LibraryByName( name );
# extent of the library
implemented_orders := lib[ 1 ];
@ -614,7 +662,7 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
# parameters for handling systematic cases, such as CCLoop( p^2, 1 )
pos_n := fail;
case := false;
if name="left Bol" or name="right Bol" then
if name="left Bol" then
divs := DivisorsInt( n );
if Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # case n = p*q
q := divs[ 2 ];
@ -633,13 +681,13 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
fi;
if name="CC" then
divs := DivisorsInt( n );
if Length( divs ) = 3 then # case p^2
if Length( divs ) = 3 and divs[ 2 ] > 7 then # case p^2, p>7
p := divs[ 2 ];
case := [p,"p^2"];
if not m in [1..3] then
Error("LOOPS: There are only 3 nonassociative CC-loops of order p^2 for an odd prime p.");
fi;
elif Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) then # p*q
elif Length( divs ) = 4 and not IsInt( divs[3]/divs[2] ) and not n=21 then # p*q
p := divs[ 3 ];
case := [p,"2*p"];
if not divs[2] = 2 then
@ -670,9 +718,6 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
if name = "left Bol" then
loop := LOOPS_ActivateLeftBolLoop( pos_n, m, case );
SetIsLeftBolLoop( loop, true );
elif name = "right Bol" then
loop := OppositeLoop( LOOPS_ActivateLeftBolLoop( pos_n, m, case ) );
SetIsRightBolLoop( loop, true );
elif name = "Moufang" then
# renaming loops so that they agree with Goodaire's classification
PG := List([1..243], i->());
@ -701,14 +746,15 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
loop := LOOPS_ActivateSteinerLoop( n, pos_n, m );
SetIsSteinerLoop( loop, true );
elif name = "CC" then
loop := LOOPS_ActivateCCLoop( n, pos_n, m, case );
if n in [2,3,5,7] then # use Cayley table for canonical cyclic group
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else
loop := LOOPS_ActivateCCLoop( n, pos_n, m, case );
fi;
SetIsCCLoop( loop, true );
elif name = "RCC" then
loop := LOOPS_ActivateRCCLoop( n, pos_n, m );
SetIsRCCLoop( loop, true );
elif name = "LCC" then
loop := OppositeLoop( LOOPS_ActivateRCCLoop( n, pos_n, m ) );
SetIsLCCLoop( loop, true );
elif name = "small" then
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
elif name = "interesting" then
@ -725,12 +771,19 @@ InstallGlobalFunction( LibraryLoop, function( name, n, m )
elif name = "nilpotent" then
loop := LOOPS_ActivateNilpotentLoop( lib[ 3 ][ pos_n ][ m ] );
elif name = "automorphic" then
if not n in [27,81,243] then # use Cayley table
if not n in [3, 9, 27, 81] then # use Cayley table
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else # use cocycles
loop := LOOPS_ActivateAutomorphicLoop( n, m );
else # use associated left Bruck loop
loop := LOOPS_ActivateAutomorphicLoop( n, m );
fi;
SetIsAutomorphicLoop( loop, true );
elif name = "right Bruck" then
if not n in [27,81] then # use Cayley table
loop := LoopByCayleyTable( LOOPS_DecodeCayleyTable( lib[ 3 ][ pos_n ][ m ] ) );
else # use cocycles
loop := LOOPS_ActivateRightBruckLoop( n, m );
fi;
SetIsRightBruckLoop( loop, true );
# up to isotopism
elif name = "itp small" then
return LibraryLoop( "small", n, lib[ 3 ][ pos_n ][ m ] );
@ -762,6 +815,8 @@ end);
#F InterestingLoop( n, m )
#F NilpotentLoop( n, m )
#F AutomorphicLoop( n, m )
#F LeftBruckLoop( n, m )
#F RightBruckLoop( n, m )
#F ItpSmallLoop( n, m )
##
@ -770,7 +825,11 @@ InstallGlobalFunction( LeftBolLoop, function( n, m )
end);
InstallGlobalFunction( RightBolLoop, function( n, m )
return LibraryLoop( "right Bol", n, m );
local loop;
loop := Opposite( LeftBolLoop( n, m ) );
SetIsRightBolLoop( loop, true );
SetName( loop, Concatenation( "<right Bol loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end);
InstallGlobalFunction( MoufangLoop, function( n, m )
@ -808,11 +867,15 @@ InstallGlobalFunction( RightConjugacyClosedLoop, function( n, m )
end);
InstallGlobalFunction( LCCLoop, function( n, m )
return LibraryLoop( "LCC", n, m );
local loop;
loop := Opposite( RCCLoop( n, m ) );
SetIsLCCLoop( loop, true );
SetName( loop, Concatenation( "<LCC loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end);
InstallGlobalFunction( LeftConjugacyClosedLoop, function( n, m )
return LibraryLoop( "LCC", n, m );
return LCCLoop( n, m );
end);
InstallGlobalFunction( SmallLoop, function( n, m )
@ -831,6 +894,18 @@ InstallGlobalFunction( AutomorphicLoop, function( n, m )
return LibraryLoop( "automorphic", n, m );
end);
InstallGlobalFunction( RightBruckLoop, function( n, m )
return LibraryLoop( "right Bruck", n, m );
end);
InstallGlobalFunction( LeftBruckLoop, function( n, m )
local loop;
loop := Opposite( RightBruckLoop( n, m ) );
SetIsLeftBruckLoop( loop, true );
SetName( loop, Concatenation( "<left Bruck loop ", String( n ), "/", String( m ), ">" ) );
return loop;
end);
InstallGlobalFunction( ItpSmallLoop, function( n, m )
return LibraryLoop( "itp small", n, m );
end);

4
gap/iso.gd
View File

@ -2,7 +2,7 @@
##
#W iso.gd Isomorphisms and isotopisms [loops]
##
#H @(#)$Id: iso.gd, v 3.2.0 2015/06/12 gap Exp $
#H @(#)$Id: iso.gd, v 3.4.0 2016/12/13 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -23,6 +23,8 @@ DeclareOperation( "IsomorphismQuasigroups", [ IsQuasigroup, IsQuasigroup ] );
DeclareOperation( "IsomorphismLoops", [ IsLoop, IsLoop ] );
DeclareOperation( "QuasigroupsUpToIsomorphism", [ IsList ] );
DeclareOperation( "LoopsUpToIsomorphism", [ IsList ] );
DeclareOperation( "QuasigroupIsomorph", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "LoopIsomorph", [ IsLoop, IsPerm ] );
DeclareOperation( "IsomorphicCopyByPerm", [ IsQuasigroup, IsPerm ] );
DeclareOperation( "IsomorphicCopyByNormalSubloop", [ IsLoop, IsLoop ] );

65
gap/iso.gi
View File

@ -2,7 +2,7 @@
##
#W iso.gi Isomorphisms and isotopisms [loops]
##
#H @(#)$Id: iso.gi, v 3.3.0 2016/10/26 gap Exp $
#H @(#)$Id: iso.gi, v 3.4.0 2017/08/24 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -466,30 +466,54 @@ end);
#############################################################################
##
#O IsomorphicCopyByPerm( Q, p )
#O QuasigroupIsomorph( Q, p )
##
## If <Q> is a quasigroup of order n and <p> a permutation of [1..n], returns
## the quasigroup (Q,*) such that p(xy) = p(x)*p(y).
## If <Q> is a loop, p is first composed with (1,1^p) to make sure
## that the neutral element of (Q,*) remains 1.
InstallMethod( IsomorphicCopyByPerm, "for a quasigroup and permutation",
InstallMethod( QuasigroupIsomorph, "for a quasigroup and permutation",
[ IsQuasigroup, IsPerm ],
function( Q, p )
local ctQ, ct, inv_p;
ctQ := CanonicalCayleyTable( CayleyTable( Q ) );
# if Q is a loop and 1^p > 1, must normalize
if (IsLoop( Q ) and (not 1^p = 1)) then
p := p * (1, 1^p );
fi;
inv_p := Inverse( p );
ct := List([1..Size(Q)], i-> List([1..Size(Q)], j ->
( ctQ[ i^inv_p ][ j^inv_p ] )^p
) );
if IsLoop( Q ) then return LoopByCayleyTable( ct ); fi;
return QuasigroupByCayleyTable( ct );
end);
#############################################################################
##
#O LoopIsomorph( Q, p )
##
## If <Q> is a loop of order n and <p> a permutation of [1..n] such that
## p(1)=1, returns the loop (Q,*) such that p(xy)=p(x)*p(y).
## If p(1)=c<>1, then the quasigroup (Q,*) is converted into loop
## via the isomorphism (1,c).
InstallMethod( LoopIsomorph, "for a loop and permutation",
[ IsLoop, IsPerm ],
function( Q, p )
return IntoLoop( QuasigroupIsomorph( Q, p ) );
end);
#############################################################################
##
#O IsomorphicCopyByPerm( Q, p )
##
## Calls LoopIsomorph( Q, p ) if <Q> is a loop,
## else QuasigroupIsotope( Q, p ).
InstallMethod( IsomorphicCopyByPerm, "for a quasigroup and permutation",
[ IsQuasigroup, IsPerm ],
function( Q, p )
if IsLoop( Q ) then
return LoopIsomorph( Q, p );
fi;
return QuasigroupIsomorph( Q, p );
end);
#############################################################################
##
#O IsomorphicCopyByNormalSubloop( L, S )
@ -594,15 +618,13 @@ end);
##
## If L1, L2 are isotopic loops, returns true, else fail.
# (MATH) First we calculate all principal loop isotopes of L1 of the form
# PrincipalLoopIsotope(L1, f, g), where f, g, are elements of L1.
# Then we filter these up to isomorphism. If L2 is isotopic to L1, then
# L2 is isomorphic to one of these principal isotopes.
# (MATH) We check for isomorphism of L2 against all principal
# isotopes of L1.
InstallMethod( IsotopismLoops, "for two loops",
[ IsLoop, IsLoop ],
function( L1, L2 )
local istps, fg, f, g, L, phi, pos, alpha, beta, gamma, p;
local f, g, L, phi, alpha, beta, gamma, p;
# make all loops canonical to be able to calculate isotopisms
if not L1 = Parent( L1 ) then L1 := LoopByCayleyTable( CayleyTable( L1 ) ); fi;
@ -619,20 +641,11 @@ function( L1, L2 )
if not Size(InnerMappingGroup(L1)) = Size(InnerMappingGroup(L2)) then return fail; fi;
# now trying to construct an isotopism
istps := [];
fg := [];
for f in L1 do for g in L1 do
Add(istps, PrincipalLoopIsotope( L1, f, g ));
Add(fg, [ f, g ] );
od; od;
for L in LoopsUpToIsomorphism( istps ) do
L := PrincipalLoopIsotope( L1, f, g );
phi := IsomorphismLoops( L, L2 );
if not phi = fail then
# must reconstruct the isotopism (alpha, beta, gamma)
# first figure out what f and g were
pos := Position( istps, L );
f := fg[ pos ][ 1 ];
g := fg[ pos ][ 2 ];
alpha := RightTranslation( L1, g );
beta := LeftTranslation( L1, f );
# we also applied an isomorphism (1,f*g) inside PrincipalLoopIsotope
@ -649,7 +662,7 @@ function( L1, L2 )
gamma := gamma * phi;
return [ alpha, beta, gamma ];
fi;
od;
od; od;
return fail;
end);

13
gap/memory.gi
View File

@ -2,7 +2,7 @@
##
#W memory.gi Memory management [loops]
##
#H @(#)$Id: memory.gi, v 3.3.0 2016/10/20 gap Exp $
#H @(#)$Id: memory.gi, v 3.4.0 2016/11/4 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -21,9 +21,14 @@ InstallGlobalFunction( LOOPS_FreeMemory, function( )
LOOPS_rcc_transitive_groups := [];
LOOPS_rcc_sections := List( [1..Length(LOOPS_rcc_data[1])], i-> [] );
LOOPS_rcc_conjugacy_classes := [ [], [] ];
# automorphic loops
LOOPS_automorphic_cocycles := [];
LOOPS_automorphic_coordinates := [];
# cc loops
LOOPS_cc_used_factors := [];
LOOPS_cc_cocycles := [];
LOOPS_cc_bases := [];
LOOPS_cc_coordinates := [];
# right Bruck loops
LOOPS_right_bruck_cocycles := [];
LOOPS_right_bruck_coordinates := [];
GASMAN("collect");
return GasmanStatistics().full.deadkb;
end);

6
gap/quasigroups.gd
View File

@ -2,7 +2,7 @@
##
#W quasigroups.gd Representing, creating and displaying quasigroups [loops]
##
#H @(#)$Id: quasigroups.gd, v 3.2.0 2016/05/02 gap Exp $
#H @(#)$Id: quasigroups.gd, v 3.4.0 2017/10/17 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -24,10 +24,10 @@ DeclareRepresentation( "IsLoopElmRep",
IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
DeclareCategory( "IsLatin", IsObject );
DeclareCategory( "IsLatinMagma", IsObject );
## quasigroup
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
DeclareCategory( "IsQuasigroup", IsMagma and IsLatinMagma );
## loop
DeclareCategory( "IsLoop", IsQuasigroup and IsMultiplicativeElementWithInverseCollection);

48
gap/quasigroups.gi
View File

@ -789,32 +789,34 @@ end );
InstallMethod( ViewObj, "for loop",
[ IsLoop ],
function( L )
if HasIsAssociative( L ) and IsAssociative( L ) then
Print( "<associative loop of order ", Size( L ), ">");
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then
Print( "<extra loop of order ", Size( L ), ">");
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then
Print( "<Moufang loop of order ", Size( L ), ">");
elif HasIsCLoop( L ) and IsCLoop( L ) then
Print( "<C loop of order ", Size( L ), ">");
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then
Print( "<left Bol loop of order ", Size( L ), ">");
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then
Print( "<right Bol loop of order ", Size( L ), ">");
elif HasIsLCLoop( L ) and IsLCLoop( L ) then
Print( "<LC loop of order ", Size( L ), ">");
elif HasIsRCLoop( L ) and IsRCLoop( L ) then
Print( "<RC loop of order ", Size( L ), ">");
local PrintMe;
PrintMe := function( name, L )
Print( "<", name, " loop of order ", Size( L ), ">");
end;
if HasIsAssociative( L ) and IsAssociative( L ) then PrintMe( "associative", L );
elif HasIsExtraLoop( L ) and IsExtraLoop( L ) then PrintMe( "extra", L );
elif HasIsMoufangLoop( L ) and IsMoufangLoop( L ) then PrintMe( "Moufang", L );
elif HasIsCLoop( L ) and IsCLoop( L ) then PrintMe( "C", L );
elif HasIsLeftBruckLoop( L ) and IsLeftBruckLoop( L ) then PrintMe( "left Bruck", L );
elif HasIsRightBruckLoop( L ) and IsRightBruckLoop( L ) then PrintMe( "right Bruck", L );
elif HasIsLeftBolLoop( L ) and IsLeftBolLoop( L ) then PrintMe( "left Bol", L );
elif HasIsRightBolLoop( L ) and IsRightBolLoop( L ) then PrintMe( "right Bol", L );
elif HasIsAutomorphicLoop( L ) and IsAutomorphicLoop( L ) then PrintMe( "automorphic", L );
elif HasIsLeftAutomorphicLoop( L ) and IsLeftAutomorphicLoop( L ) then PrintMe( "left automorphic", L );
elif HasIsRightAutomorphicLoop( L ) and IsRightAutomorphicLoop( L ) then PrintMe( "right automorphic", L );
elif HasIsLCLoop( L ) and IsLCLoop( L ) then PrintMe( "LC", L );
elif HasIsRCLoop( L ) and IsRCLoop( L ) then PrintMe( "RC", L );
elif HasIsLeftAlternative( L ) and IsLeftAlternative( L ) then
if HasIsRightAlternative( L ) and IsRightAlternative( L ) then
Print( "<alternative loop of order ", Size( L ), ">");
else
Print( "<left alternative loop of order ", Size( L ), ">");
PrintMe("alternative", L );
else
PrintMe("left alternative", L );
fi;
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then
Print( "<right alternative loop of order ", Size( L ), ">");
elif HasIsFlexible( L ) and IsFlexible( L ) then
Print( "<flexible loop of order ", Size( L ), ">");
elif HasIsRightAlternative( L ) and IsRightAlternative( L ) then PrintMe( "right alternative", L );
elif HasIsCommutative( L ) and IsCommutative( L ) then PrintMe( "commutative", L );
elif HasIsFlexible( L ) and IsFlexible( L ) then PrintMe( "flexible", L);
else
# MORE ??
Print( "<loop of order ", Size( L ), ">" );

2
tst/bol.tst
View File

@ -19,7 +19,7 @@ gap> IsomorphismLoops(B,LeftBolLoop(15,1));
gap> Q := RightBolLoop(15,1);;
gap> AssociatedRightBruckLoop( Q );
<right Bol loop of order 15>
<right Bruck loop of order 15>
# TESTING EXACT GROUP FACTORIZATIONS

2
tst/core_methods.tst
View File

@ -2,7 +2,7 @@
##
#W core_methods.tst Testing core methods G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: core_methods.tst, v 3.3.0 2016/10/26 gap Exp $
#H @(#)$Id: core_methods.tst, v 3.4.0 2017/10/26 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)

7
tst/iso.tst
View File

@ -2,7 +2,7 @@
##
#W iso.tst Testing isomorphisms G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: iso.tst, v 3.2.0 2016/06/02 gap Exp $
#H @(#)$Id: iso.tst, v 3.4.0 2017/10/26 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -34,6 +34,11 @@ Group([ (1,2,3), (1,3,2) ])
gap> Q := DirectProduct( MoufangLoop( 32, 5 ) );;
gap> Qp := IsomorphicCopyByPerm( Q, (2,3,4)(17,20) );;
gap> Qq := LoopIsomorph( Q, (2,3,4)(17,20) );;
gap> Qp = Qq;
false
gap> CayleyTable( Qp ) = CayleyTable( Qq );
true
gap> IsomorphismLoops( Q, Qp );
(2,3,4)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)
gap> LoopsUpToIsomorphism( [Q,Qp] );

54
tst/lib.tst
View File

@ -2,7 +2,7 @@
##
#W lib.tst Testing libraries of loops G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: lib.tst, v 3.3.0 2016/10/26 gap Exp $
#H @(#)$Id: lib.tst, v 3.4.0 2017/10/26 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
@ -156,19 +156,34 @@ gap> LCCLoop(6,3); LCCLoop(25,119);
# CC LOOPS
gap> DisplayLibraryInfo("CC");
The library contains all nonassociative CC loops of order less than 28
The library contains all CC loops of order
2<=2^k<=64, 3<=3^k<=81, 5<=5^k<=125, 7<=7^k<=343,
all nonassociative CC loops of order less than 28,
and all nonassociative CC loops of order p^2 and 2*p for any odd prime p.
------
Extent of the library:
2 loops of order 8
1 loop of order 2
1 loop of order 3
2 loops of order 4
1 loop of order 5
1 loop of order 7
7 loops of order 8
5 loops of order 9
3 loops of order 12
28 loops of order 16
42 loops of order 16
7 loops of order 18
3 loops of order 20
1 loop of order 21
14 loops of order 24
55 loops of order 27
3 loops of order p^2 for every odd prime p,
5 loops of order 25
60 loops of order 27
437 loops of order 32
5 loops of order 49
14854 loops of order 64
5406 loops of order 81
84 loops of order 125
122 loops of order 343
3 loops of order p^2 for every prime p>7,
1 loop of order 2*p for every odd prime p
true
@ -233,10 +248,7 @@ gap> CodeLoop( 64, 80 );
gap> DisplayLibraryInfo("automorphic");
The library contains:
- all nonassociative automorphic loops of order less than 16,
- all commutative automorphic loops of order 3, 9, 27, 81,
- all commutative automorphic loops of order 243 that are central
extensions of Z_3 by F, where F is not the elem. ab. 3-group.
Note: Abelian groups are included among the commutative loops.
- all commutative automorphic loops of order 3, 9, 27, 81.
------
Extent of the library:
1 loop of order 3
@ -249,7 +261,6 @@ Extent of the library:
2 loops of order 15
7 loops of order 27
72 loops of order 81
118451 loops of order 243
true
gap> AutomorphicLoop(15,2);
@ -258,7 +269,24 @@ gap> AutomorphicLoop(15,2);
gap> AutomorphicLoop(27,1);
<automorphic loop 27/1>
gap> AutomorphicLoop(243,100);
<automorphic loop 243/100>
gap> AutomorphicLoop(81,10);
<automorphic loop 81/10>
# RIGHT BRUCK LOOPS
gap> DisplayLibraryInfo("right Bruck");
The library contains all right Bruck loops of orders 3, 9, 27 and 81.
------
Extent of the library:
1 loop of order 3
2 loops of order 9
7 loops of order 27
72 loops of order 81
true
gap> RightBruckLoop(81,3);
<right Bruck loop 81/3>
gap> STOP_TEST( "lib.tst", 10000000 );

12
tst/testall.g
View File

@ -2,15 +2,15 @@
##
#W testall.g Testing LOOPS G. P. Nagy / P. Vojtechovsky
##
#H @(#)$Id: testall.g, v 3.0.0 2015/06/15 gap Exp $
#H @(#)$Id: testall.g, v 3.4.0 2017/10/26 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
dirs := DirectoriesPackageLibrary( "loops", "tst" );
ReadTest( Filename( dirs, "core_methods.tst" ) );
ReadTest( Filename( dirs, "nilpot.tst" ) );
ReadTest( Filename( dirs, "iso.tst" ) );
ReadTest( Filename( dirs, "lib.tst" ) );
ReadTest( Filename( dirs, "bol.tst" ) );
Test( Filename( dirs, "core_methods.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "nilpot.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "iso.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "lib.tst" ), rec( compareFunction := "uptowhitespace" ) );
Test( Filename( dirs, "bol.tst" ), rec( compareFunction := "uptowhitespace" ) );