update to LOOPS 3.4.0
These are simply the changes as distributed.
This commit is contained in:
Glen Whitney
parent
7e8b3b5562
commit
f64208f12f
@ -1,9 +1,9 @@
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SetPackageInfo( rec(
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PackageName := "loops",
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Subtitle := "Computing with quasigroups and loops in GAP",
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Version := "3.3.0",
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Date := "26/10/2016",
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ArchiveURL := "http://www.math.du.edu/loops/loops-3.3.0",
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Version := "3.4.0",
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Date := "27/10/2017",
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ArchiveURL := "http://www.math.du.edu/loops/loops-3.4.0",
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ArchiveFormats := "-win.zip .tar.gz",
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Persons := [
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@ -83,7 +83,7 @@ Dependencies := rec(
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),
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AvailabilityTest := ReturnTrue,
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BannerString := "This version of LOOPS is ready for GAP 4.7.\n",
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BannerString := "This version of LOOPS is ready for GAP 4.8.\n",
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Autoload := false, # false for deposited packages
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TestFile := "tst/testall.g",
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@ -2,39 +2,21 @@
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##
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#W automorphic.tbl Automorphic loops G. P. Nagy / P. Vojtechovsky
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##
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#H @(#)$Id: automorphic.tbl, v 3.3.0 2016/10/20 gap Exp $
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#H @(#)$Id: automorphic.tbl, v 3.4.0 2017/10/23 gap Exp $
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##
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#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
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#Y P. Vojtechovsky (University of Denver, USA)
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##
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#############################################################################
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## Binding global variables
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## LOOPS_automorphic_cocycles
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## LOOPS_automorphic_bases
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## LOOPS_automorphic_coordinates
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# Many small automorphic loops are represtented by encoded Cayley tables.
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#
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# Commutative automorphic loops of order 243 are represtented as central
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# extensions of the cyclic group of order 3.
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# The necessary data is only loaded on demand and consists of:
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# - LOOPS_automorphic_cocycles, a list of encoded bases of the
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# space of cocycles modulo coboundaries for every factor loop F needed.
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# - LOOPS_automorphic_coordinates, a list that for every loop
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# points to the factor loop and gives coordinates of the required cocycle
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# with respect to the relevant basis.
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LOOPS_automorphic_data := [
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#implemented orders
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[3,6,8,9,10,12,14,15,27,81,243],
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[3,6,8,9,10,12,14,15,27,81],
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#number of nonassociative loops of given order
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[1,1,7,2,3,2,5,2,7,72,118451],
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[1,1,7,2,3,2,5,2,7,72],
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#the loops
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[
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#order 3 (Z_3)
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#order 3 (Z_3, use left Bruck loops, placeholder only)
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[
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"201"
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],
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#order 6
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[
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@ -50,10 +32,8 @@ LOOPS_automorphic_data := [
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"0325476301675421076455760132467102374523106543201",
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"0325476310674520176545761023467013275432106452301"
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],
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#order 9 (two abelian groups)
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#order 9 (two abelian groups, use left Bruck loops, placeholder only)
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[
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"204537861534867678012861207201345534",
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"204537861534867678120862017012453345"
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]
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,
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#order 10
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@ -80,20 +60,13 @@ LOOPS_automorphic_data := [
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"234068597BDAEC340189675DEBCA401297856ECDAB012375968CAEBD6897ADECB041328975DCABE430215689EABDC102439756CBDEA324107568BECAD21304BDEC0413258976DECA4302187569ABDE1024395687ECAB3241076895CABD2130469758",
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"234067895BCDEA340178956CDEAB401289567DEABC012395678EABCD7968ADBEC012348579ECADB340129685DBECA123405796CADBE401236857BECAD23401DBEC0432156789ECAD3210478956ADBE1043295678BECA4321067895CADB2104389567"
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],
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#order 27 (commutative only, placeholder)
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#order 27 (commutative only, use left Bruck loops, placeholder)
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[
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]
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,
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#order 81 (commutative only, placeholder)
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[
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]
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,
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#order 243 (commutative only, placeholder)
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#order 81 (commutative only, use left Bruck loops, placeholder)
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[
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]
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]
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];
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LOOPS_automorphic_cocycles := [];
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LOOPS_automorphic_bases := [];
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LOOPS_automorphic_coordinates := [];
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File diff suppressed because it is too large
Load Diff
61
data/cc.tbl
61
data/cc.tbl
@ -1,31 +1,52 @@
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#############################################################################
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##
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#W cc.tbl CC-loops p^2, 2p, for p odd prime G. P. Nagy / P. Vojtechovsky
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#W cc.tbl Library of CC loops G. P. Nagy / P. Vojtechovsky
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##
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#H @(#)$Id: cc.tbl, v 3.0.0 2015/06/10 gap Exp $
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#H @(#)$Id: cc.tbl, v 3.4.0 2015/06/10 gap Exp $
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##
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#Y Copyright (C) 2005, G. P. Nagy (University of Szeged, Hungary),
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#Y P. Vojtechovsky (University of Denver, USA)
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##
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#############################################################################
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## Binding global variables
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## LOOPS_cc_used_factors
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## LOOPS_cc_cocycles
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## LOOPS_cc_bases
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## LOOPS_cc_coordinates
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# CC loops are activated as follows:
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# If n = 2p or p^2, where p is a prime, then we call a method for
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# cosntructing these loops.
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# If n = 2p, where p is an odd prime, then we call an algebraic method for
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# constructing these loops.
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# If n = p^2, where p>3 is a prime, then we call an algebraic method for
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# construction these loops.
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# If n is a power of 2 or 3, then we use cocycles located in cc/cc_cocycles_n.tbl.
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# For all other orders, we point to the library of RCC loops.
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LOOPS_cc_data := [
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#implemented orders
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[ 8, 12, 16, 18, 20, 21, 24, 27],
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#number of nonassociative loops of given order
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[ 2, 3, 28, 7, 3, 1, 14, 55],
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#the numbers of the loops in the RCC library
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[ 2, 3, 4, 5, 7, 8, 9, 12, 16, 18, 20, 21, 24, 25, 27, 32, 49, 64, 81, 125, 343],
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#number of loops of given order in the library
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[ 1, 1, 2, 1, 1, 7, 5, 3, 42, 7, 3, 1, 14, 5, 60, 437, 5, 14854, 5406, 84, 122],
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[
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#order 8
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[2,7],
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#order 2 (Z_2)
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["010"],
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#order 3 (Z_3)
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["201"],
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#order 4 (placeholder only)
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,
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# order 5 (Z_5)
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["2340401123"],
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# order 7 (Z_7)
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["234560456016012123345"],
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#order 8 (placeholder only)
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,
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#order 9 (placeholder only)
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,
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#order 12
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[53,73,89],
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#order 16
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[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],
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#order 16 (placeholder only)
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,
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#order 18
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[22,29,77,292,360,377,1133],
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#order 20
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@ -33,18 +54,12 @@ LOOPS_cc_data := [
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#order 21
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[104],
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#order 24
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[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216],
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#order 27
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[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]
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[302,1025,2119,2182,2335,3066,4569,5176,5589,5997,7495,194830,225705,243216]
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]
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];
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# The following can be used to point to CC loops of order 2p and p^2 in the library of RCC loops.
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# order 6, [3]
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# order 9, [5,4,3]
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# order 10, [16]
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# order 14, [97]
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# order 22, [10346]
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# order 25, [86,93,118]
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# order 26, [151964]
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LOOPS_cc_used_factors := [];
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LOOPS_cc_cocycles := [];
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LOOPS_cc_bases := [];
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LOOPS_cc_coordinates := [];
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@ -6,7 +6,7 @@
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[1XComputing with quasigroups and loops in [5XGAP[105X[101X
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Version 3.3.0
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Version 3.4.0
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Gábor P. Nagy
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@ -28,7 +28,7 @@
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-------------------------------------------------------
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[1XCopyright[101X
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[33X[0;0Y© 2016 Gábor P. Nagy and Petr Vojtěchovský.[133X
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[33X[0;0Y© 2017 Gábor P. Nagy and Petr Vojtěchovský.[133X
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-------------------------------------------------------
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@ -167,10 +167,12 @@
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6.11-3 QuasigroupsUpToIsomorphism
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6.11-4 LoopsUpToIsomorphism
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6.11-5 AutomorphismGroup
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6.11-6 IsomorphicCopyByPerm
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6.11-7 IsomorphicCopyByNormalSubloop
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6.11-8 Discriminator
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6.11-9 AreEqualDiscriminators
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6.11-6 QuasigroupIsomorph
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6.11-7 LoopIsomorph
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6.11-8 IsomorphicCopyByPerm
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6.11-9 IsomorphicCopyByNormalSubloop
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6.11-10 Discriminator
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6.11-11 AreEqualDiscriminators
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6.12 [33X[0;0YIsotopisms[133X
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6.12-1 IsotopismLoops
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6.12-2 LoopsUpToIsotopism
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@ -256,28 +258,31 @@
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9.2 [33X[0;0YLeft Bol Loops and Right Bol Loops[133X
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9.2-1 LeftBolLoop
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9.2-2 RightBolLoop
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9.3 [33X[0;0YMoufang Loops[133X
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9.3-1 MoufangLoop
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9.4 [33X[0;0YCode Loops[133X
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9.4-1 CodeLoop
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9.5 [33X[0;0YSteiner Loops[133X
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9.5-1 SteinerLoop
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9.6 [33X[0;0YConjugacy Closed Loops[133X
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9.6-1 [33X[0;0YRCCLoop and RightConjugacyClosedLoop[133X
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9.6-2 [33X[0;0YLCCLoop and LeftConjugacyClosedLoop[133X
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9.6-3 [33X[0;0YCCLoop and ConjugacyClosedLoop[133X
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9.7 [33X[0;0YSmall Loops[133X
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9.7-1 SmallLoop
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9.8 [33X[0;0YPaige Loops[133X
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9.8-1 PaigeLoop
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9.9 [33X[0;0YNilpotent Loops[133X
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9.9-1 NilpotentLoop
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9.10 [33X[0;0YAutomorphic Loops[133X
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9.10-1 AutomorphicLoop
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9.11 [33X[0;0YInteresting Loops[133X
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9.11-1 InterestingLoop
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9.12 [33X[0;0YLibraries of Loops Up To Isotopism[133X
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9.12-1 ItpSmallLoop
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9.3 [33X[0;0YLeft Bruck Loops and Right Bruck Loops[133X
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9.3-1 LeftBruckLoop
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9.3-2 RightBruckLoop
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9.4 [33X[0;0YMoufang Loops[133X
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9.4-1 MoufangLoop
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9.5 [33X[0;0YCode Loops[133X
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9.5-1 CodeLoop
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9.6 [33X[0;0YSteiner Loops[133X
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9.6-1 SteinerLoop
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9.7 [33X[0;0YConjugacy Closed Loops[133X
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9.7-1 [33X[0;0YRCCLoop and RightConjugacyClosedLoop[133X
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9.7-2 [33X[0;0YLCCLoop and LeftConjugacyClosedLoop[133X
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9.7-3 [33X[0;0YCCLoop and ConjugacyClosedLoop[133X
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9.8 [33X[0;0YSmall Loops[133X
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9.8-1 SmallLoop
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9.9 [33X[0;0YPaige Loops[133X
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9.9-1 PaigeLoop
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9.10 [33X[0;0YNilpotent Loops[133X
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9.10-1 NilpotentLoop
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9.11 [33X[0;0YAutomorphic Loops[133X
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9.11-1 AutomorphicLoop
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9.12 [33X[0;0YInteresting Loops[133X
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9.12-1 InterestingLoop
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9.13 [33X[0;0YLibraries of Loops Up To Isotopism[133X
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9.13-1 ItpSmallLoop
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A [33X[0;0YFiles[133X
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B [33X[0;0YFilters[133X
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@ -31,7 +31,7 @@
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<h2>Computing with quasigroups and loops in <strong class="pkg">GAP</strong></h2>
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<p>Version 3.3.0</p>
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<p>Version 3.4.0</p>
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</div>
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<p><b>Gábor P. Nagy
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@ -50,7 +50,7 @@
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<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
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<h3>Copyright</h3>
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<p>© 2016 Gábor P. Nagy and Petr Vojtěchovský.</p>
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<p>© 2017 Gábor P. Nagy and Petr Vojtěchovský.</p>
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<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
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@ -331,10 +331,12 @@
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
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</span>
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@ -474,60 +476,66 @@
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7953702D84E60AF4">9.3 <span class="Heading">Moufang Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X8028D69A86B15897">9.3 <span class="Heading">Left Bruck Loops and Right Bruck Loops</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
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</div></div>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7BCA6BCB847F79DC">9.4 <span class="Heading">Code Loops</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9_mj.html#X7953702D84E60AF4">9.4 <span class="Heading">Moufang Loops</span></a>
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</span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
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</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </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"> </span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </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>
|
||||
|
||||
@ -20,7 +20,7 @@
|
||||
|
||||
[1X1.2 [33X[0;0YInstallation[133X[101X
|
||||
|
||||
[33X[0;0YHave [5XGAP 4.7[105X or newer installed on your computer.[133X
|
||||
[33X[0;0YHave [5XGAP 4.8[105X or newer installed on your computer.[133X
|
||||
|
||||
[33X[0;0YIf you do not see the subfolder [11Xpkg/loops[111X in the main directory of [5XGAP[105X then
|
||||
download the [5XLOOPS[105X package from the distribution website
|
||||
@ -85,14 +85,15 @@
|
||||
|
||||
[33X[0;0YWe 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.[133X
|
||||
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.[133X
|
||||
|
||||
[33X[0;0YThe 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.[133X
|
||||
Katharina Artic. The library of right Bruck loops of order 27, 81 was
|
||||
obtained jointly with Izabella Stuhl.[133X
|
||||
|
||||
[33X[0;0YGábor P. Nagy was supported by OTKA grants F042959 and T043758, and Petr
|
||||
Vojtěchovský was supported by the 2006 and 2016 University of Denver PROF
|
||||
|
||||
@ -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> 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>
|
||||
|
||||
|
||||
@ -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);
|
||||
|
||||
|
||||
@ -81,8 +81,8 @@ 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);
|
||||
|
||||
|
||||
@ -488,17 +488,28 @@
|
||||
the elements of the underlying quasigroup without changing the isomorphism
|
||||
type of the quasigroups. [5XLOOPS[105X contains several functions for this purpose.[133X
|
||||
|
||||
[1X6.11-6 IsomorphicCopyByPerm[101X
|
||||
[1X6.11-6 QuasigroupIsomorph[101X
|
||||
|
||||
[29X[2XQuasigroupIsomorph[102X( [3XQ[103X, [3Xf[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10YWhen [3XQ[103X is a quasigroup and [3Xf[103X is a permutation of [22X1,dots,|[122X[3XQ[103X[22X|[122X,
|
||||
returns the quasigroup defined on the same set as [3XQ[103X with
|
||||
multiplication [22X*[122X defined by [22Xx*y =[122X[3Xf[103X[22X([122X[3Xf[103X[22X^-1(x)[122X[3Xf[103X[22X^-1(y))[122X.[133X
|
||||
|
||||
[1X6.11-7 LoopIsomorph[101X
|
||||
|
||||
[29X[2XLoopIsomorph[102X( [3XQ[103X, [3Xf[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10YWhen [3XQ[103X is a loop and [3Xf[103X is a permutation of [22X1,dots,|[122X[3XQ[103X[22X|[122X fixing [22X1[122X,
|
||||
returns the loop defined on the same set as [3XQ[103X with multiplication
|
||||
[22X*[122X defined by [22Xx*y =[122X[3Xf[103X[22X([122X[3Xf[103X[22X^-1(x)[122X[3Xf[103X[22X^-1(y))[122X. If [3Xf[103X[22X(1)=cne 1[122X, the
|
||||
isomorphism [22X(1,c)[122X is applied after [3Xf[103X.[133X
|
||||
|
||||
[1X6.11-8 IsomorphicCopyByPerm[101X
|
||||
|
||||
[29X[2XIsomorphicCopyByPerm[102X( [3XQ[103X, [3Xf[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10YWhen [3XQ[103X is a quasigroup and [3Xf[103X is a permutation of [22X1,dots,|[122X[3XQ[103X[22X|[122X,
|
||||
returns a quasigroup defined on the same set as [3XQ[103X with
|
||||
multiplication [22X*[122X defined by [22Xx*y =[122X[3Xf[103X[22X([122X[3Xf[103X[22X^-1(x)[122X[3Xf[103X[22X^-1(y))[122X. When [3XQ[103X is a
|
||||
declared loop, a loop is returned. Consequently, when [3XQ[103X is a
|
||||
declared loop and [3Xf[103X[22X(1) = kne 1[122X, then [3Xf[103X is first replaced with [3Xf[103X[22X∘
|
||||
(1,k)[122X, to make sure that the resulting Cayley table is normalized.[133X
|
||||
[6XReturns:[106X [33X[0;10Y[10XLoopIsomorphism([3XQ[103X[10X,[3Xf[103X[10X)[110X if [3XQ[103X is a loop, and
|
||||
[10XQuasigroupIsomorphism([3XQ[103X[10X,[3Xf[103X[10X)[110X if [3XQ[103X is a quasigroup.[133X
|
||||
|
||||
[1X6.11-7 IsomorphicCopyByNormalSubloop[101X
|
||||
[1X6.11-9 IsomorphicCopyByNormalSubloop[101X
|
||||
|
||||
[29X[2XIsomorphicCopyByNormalSubloop[102X( [3XQ[103X, [3XS[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10YWhen [3XS[103X is a normal subloop of a loop [3XQ[103X, 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.[133X
|
||||
|
||||
[1X6.11-8 Discriminator[101X
|
||||
[1X6.11-10 Discriminator[101X
|
||||
|
||||
[29X[2XDiscriminator[102X( [3XQ[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10YA data structure with isomorphism invariants of a loop [3XQ[103X.[133X
|
||||
@ -523,7 +534,7 @@
|
||||
[33X[0;0YIf two loops have different discriminators, they are not isomorphic. If they
|
||||
have identical discriminators, they may or may not be isomorphic.[133X
|
||||
|
||||
[1X6.11-9 AreEqualDiscriminators[101X
|
||||
[1X6.11-11 AreEqualDiscriminators[101X
|
||||
|
||||
[29X[2XAreEqualDiscriminators[102X( [3XD1[103X, [3XD2[103X ) [32X operation
|
||||
[6XReturns:[106X [33X[0;10Y[10Xtrue[110X if [3XD1[103X, [3XD2[103X are equal discriminators for the purposes of
|
||||
|
||||
@ -120,10 +120,12 @@
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X82373C5479574F22">6.11-3 QuasigroupsUpToIsomorphism</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8308F38283C61B20">6.11-4 LoopsUpToIsomorphism</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X87677B0787B4461A">6.11-5 AutomorphismGroup</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-6 IsomorphicCopyByPerm</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-7 IsomorphicCopyByNormalSubloop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-8 Discriminator</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-9 AreEqualDiscriminators</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7A42812B7B027DD4">6.11-6 QuasigroupIsomorph</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7BD1AC32851286EA">6.11-7 LoopIsomorph</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X85B3E22679FD8D81">6.11-8 IsomorphicCopyByPerm</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X8121DE3A78795040">6.11-9 IsomorphicCopyByNormalSubloop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X7D09D8957E4A0973">6.11-10 Discriminator</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6_mj.html#X812F0DEE7C896E18">6.11-11 AreEqualDiscriminators</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7E996BDD81E594F9">6.12 <span class="Heading">Isotopisms</span></a>
|
||||
</span>
|
||||
@ -607,16 +609,30 @@ MappingByFunction( <loop of order 12>, <loop of order 4>,
|
||||
|
||||
<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">‣ 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">‣ 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">‣ 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">‣ 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( <loop of order 12>, <loop of order 4>,
|
||||
|
||||
<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">‣ 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( <loop of order 12>, <loop of order 4>,
|
||||
|
||||
<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">‣ 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>
|
||||
|
||||
@ -78,12 +78,34 @@
|
||||
retrieved by calling [10XOpposite[110X on left Bol loops.[133X
|
||||
|
||||
|
||||
[1X9.3 [33X[0;0YMoufang Loops[133X[101X
|
||||
[1X9.3 [33X[0;0YLeft Bruck Loops and Right Bruck Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe emmerging library named [13Xleft Bruck[113X contains all left Bruck loops of
|
||||
orders [22X3[122X, [22X9[122X, [22X27[122X and [22X81[122X (there are [22X1[122X, [22X2[122X, [22X7[122X and [22X72[122X such loops, respectively).[133X
|
||||
|
||||
[33X[0;0YFor an odd prime [22Xp[122X, left Bruck loops of order [22Xp^k[122X are centrally nilpotent
|
||||
and hence central extensions of the cyclic group of order [22Xp[122X by a left Bruck
|
||||
loop of order [22Xp^k-1[122X. It is known that left Bruck loops of order [22Xp[122X and [22Xp^2[122X
|
||||
are abelian groups; we have included them in the library because of the
|
||||
iterative nature of the construction of nilpotent loops.[133X
|
||||
|
||||
[1X9.3-1 LeftBruckLoop[101X
|
||||
|
||||
[29X[2XLeftBruckLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth left Bruck loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
[1X9.3-2 RightBruckLoop[101X
|
||||
|
||||
[29X[2XRightBruckLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth right Bruck loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.4 [33X[0;0YMoufang Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13XMoufang[113X contains all nonassociative Moufang loops of order
|
||||
[22Xnle 64[122X and [22Xn∈{81,243}[122X.[133X
|
||||
|
||||
[1X9.3-1 MoufangLoop[101X
|
||||
[1X9.4-1 MoufangLoop[101X
|
||||
|
||||
[29X[2XMoufangLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth Moufang loop of order [3Xn[103X in the library.[133X
|
||||
@ -107,20 +129,20 @@
|
||||
obtained as [10XMoufangLoop(16,3)[110X.[133X
|
||||
|
||||
|
||||
[1X9.4 [33X[0;0YCode Loops[133X[101X
|
||||
[1X9.5 [33X[0;0YCode Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13Xcode[113X 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.[133X
|
||||
|
||||
[1X9.4-1 CodeLoop[101X
|
||||
[1X9.5-1 CodeLoop[101X
|
||||
|
||||
[29X[2XCodeLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth code loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.5 [33X[0;0YSteiner Loops[133X[101X
|
||||
[1X9.6 [33X[0;0YSteiner Loops[133X[101X
|
||||
|
||||
[33X[0;0YHere is how the libary named [13XSteiner[113X is described within [5XLOOPS[105X:[133X
|
||||
|
||||
@ -141,13 +163,13 @@
|
||||
[33X[0;0YOur labeling of Steiner loops of order 16 coincides with the labeling of
|
||||
Steiner triple systems of order 15 in [CR99].[133X
|
||||
|
||||
[1X9.5-1 SteinerLoop[101X
|
||||
[1X9.6-1 SteinerLoop[101X
|
||||
|
||||
[29X[2XSteinerLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth Steiner loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.6 [33X[0;0YConjugacy Closed Loops[133X[101X
|
||||
[1X9.7 [33X[0;0YConjugacy Closed Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13XRCC[113X contains all nonassocitive right conjugacy closed
|
||||
loops of order [22Xnle 27[122X up to isomorphism. The data for the library was
|
||||
@ -171,14 +193,14 @@
|
||||
|
||||
|
||||
|
||||
[1X9.6-1 [33X[0;0YRCCLoop and RightConjugacyClosedLoop[133X[101X
|
||||
[1X9.7-1 [33X[0;0YRCCLoop and RightConjugacyClosedLoop[133X[101X
|
||||
|
||||
[29X[2XRCCLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[29X[2XRightConjugacyClosedLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth right conjugacy closed loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.6-2 [33X[0;0YLCCLoop and LeftConjugacyClosedLoop[133X[101X
|
||||
[1X9.7-2 [33X[0;0YLCCLoop and LeftConjugacyClosedLoop[133X[101X
|
||||
|
||||
[29X[2XLCCLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[29X[2XLeftConjugacyClosedLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
@ -188,8 +210,10 @@
|
||||
Left conjugacy closed loops are obtained from right conjugacy closed loops
|
||||
via [10XOpposite[110X.[133X
|
||||
|
||||
[33X[0;0YThe library named [13XCC[113X contains all nonassociative conjugacy closed loops of
|
||||
order [22Xnle 27[122X and also of orders [22X2p[122X and [22Xp^2[122X for all primes [22Xp[122X.[133X
|
||||
[33X[0;0YThe library named [13XCC[113X contains all CC loops of order [22X2le 2^kle 64[122X, [22X3le 3^kle
|
||||
81[122X, [22X5le 5^kle 125[122X, [22X7le 7^kle 343[122X, all nonassociative CC loops of order less
|
||||
than 28, and all nonassociative CC loops of order [22Xp^2[122X and [22X2p[122X for any odd
|
||||
prime [22Xp[122X.[133X
|
||||
|
||||
[33X[0;0YBy results of Kunen [Kun00], for every odd prime [22Xp[122X there are precisely 3
|
||||
nonassociative conjugacy closed loops of order [22Xp^2[122X. Csörgő and Drápal [CD05]
|
||||
@ -215,25 +239,25 @@
|
||||
m + n )[122X.[133X
|
||||
|
||||
|
||||
[1X9.6-3 [33X[0;0YCCLoop and ConjugacyClosedLoop[133X[101X
|
||||
[1X9.7-3 [33X[0;0YCCLoop and ConjugacyClosedLoop[133X[101X
|
||||
|
||||
[29X[2XCCLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[29X[2XConjugacyClosedLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth conjugacy closed loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.7 [33X[0;0YSmall Loops[133X[101X
|
||||
[1X9.8 [33X[0;0YSmall Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13Xsmall[113X contains all nonassociative loops of order 5 and 6.
|
||||
There are 5 and 107 such loops, respectively.[133X
|
||||
|
||||
[1X9.7-1 SmallLoop[101X
|
||||
[1X9.8-1 SmallLoop[101X
|
||||
|
||||
[29X[2XSmallLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.8 [33X[0;0YPaige Loops[133X[101X
|
||||
[1X9.9 [33X[0;0YPaige Loops[133X[101X
|
||||
|
||||
[33X[0;0Y[13XPaige loops[113X are nonassociative finite simple Moufang loops. By [Lie87],
|
||||
there is precisely one Paige loop for every finite field.[133X
|
||||
@ -241,14 +265,14 @@
|
||||
[33X[0;0YThe library named [13XPaige[113X contains the smallest nonassociative simple Moufang
|
||||
loop.[133X
|
||||
|
||||
[1X9.8-1 PaigeLoop[101X
|
||||
[1X9.9-1 PaigeLoop[101X
|
||||
|
||||
[29X[2XPaigeLoop[102X( [3Xq[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe Paige loop constructed over the finite field of order [3Xq[103X. Only
|
||||
the case [10X[3Xq[103X[10X=2[110X is implemented.[133X
|
||||
|
||||
|
||||
[1X9.9 [33X[0;0YNilpotent Loops[133X[101X
|
||||
[1X9.10 [33X[0;0YNilpotent Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13Xnilpotent[113X 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.[133X
|
||||
|
||||
[1X9.9-1 NilpotentLoop[101X
|
||||
[1X9.10-1 NilpotentLoop[101X
|
||||
|
||||
[29X[2XNilpotentLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth nilpotent loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.10 [33X[0;0YAutomorphic Loops[133X[101X
|
||||
[1X9.11 [33X[0;0YAutomorphic Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13Xautomorphic[113X 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 [22XQ[122X of order 243 possessing a central subloop [22XS[122X
|
||||
of order 3 such that [22XQ/S[122X is not the elementary abelian group of order 81
|
||||
(there are 118451 such loops).[133X
|
||||
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).[133X
|
||||
|
||||
[1X9.10-1 AutomorphicLoop[101X
|
||||
[33X[0;0YIt 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.[133X
|
||||
|
||||
[1X9.11-1 AutomorphicLoop[101X
|
||||
|
||||
[29X[2XAutomorphicLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth automorphic loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.11 [33X[0;0YInteresting Loops[133X[101X
|
||||
[1X9.12 [33X[0;0YInteresting Loops[133X[101X
|
||||
|
||||
[33X[0;0YThe library named [13Xinteresting[113X 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.[133X
|
||||
|
||||
[1X9.11-1 InterestingLoop[101X
|
||||
[1X9.12-1 InterestingLoop[101X
|
||||
|
||||
[29X[2XInterestingLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth interesting loop of order [3Xn[103X in the library.[133X
|
||||
|
||||
|
||||
[1X9.12 [33X[0;0YLibraries of Loops Up To Isotopism[133X[101X
|
||||
[1X9.13 [33X[0;0YLibraries of Loops Up To Isotopism[133X[101X
|
||||
|
||||
[33X[0;0YFor the library named [13Xsmall[113X we also provide the corresponding library of
|
||||
loops up to isotopism. In general, given a library named [13Xlibname[113X, the
|
||||
corresponding library of loops up to isotopism is named [13Xitp lib[113X, and the
|
||||
loops can be retrieved by the template [10XItpLibLoop(n,m)[110X.[133X
|
||||
|
||||
[1X9.12-1 ItpSmallLoop[101X
|
||||
[1X9.13-1 ItpSmallLoop[101X
|
||||
|
||||
[29X[2XItpSmallLoop[102X( [3Xn[103X, [3Xm[103X ) [32X function
|
||||
[6XReturns:[106X [33X[0;10YThe [3Xm[103Xth small loop of order [3Xn[103X up to isotopism in the library.[133X
|
||||
|
||||
@ -38,60 +38,66 @@
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7EE99F647C537994">9.2-1 LeftBolLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8774304282654C58">9.2-2 RightBolLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X81E82098822543EE">9.3-1 MoufangLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X8290B01780F0FCD3">9.3-1 LeftBruckLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X798DD7CF871F648F">9.3-2 RightBruckLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.4-1 CodeLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X81E82098822543EE">9.4-1 MoufangLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X87C235457E859AF4">9.5-1 SteinerLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7DB4D3B27BB4D7EE">9.5-1 CodeLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </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"> </span><a href="chap9_mj.html#X87C235457E859AF4">9.6-1 SteinerLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </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"> </span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.7-1 SmallLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7C6EE23E84CD87D3">9.8-1 SmallLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.8-1 PaigeLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7FCF4D6B7AD66D74">9.9-1 PaigeLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.9-1 NilpotentLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X7A9C960D86E2AD28">9.10-1 NilpotentLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.10-1 AutomorphicLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X784FFA9E7FDA9F43">9.11-1 AutomorphicLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X87F24AD3811910D3">9.11-1 InterestingLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X87F24AD3811910D3">9.12-1 InterestingLoop</a></span>
|
||||
</div></div>
|
||||
<div class="ContSect"><span class="tocline"><span class="nocss"> </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"> </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"> </span><a href="chap9_mj.html#X850C4C01817A098D">9.12-1 ItpSmallLoop</a></span>
|
||||
<span class="ContSS"><br /><span class="nocss"> </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">‣ 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">‣ 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">‣ 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>
|
||||
@ -198,20 +226,20 @@
|
||||
|
||||
<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">‣ 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>
|
||||
|
||||
@ -234,14 +262,14 @@ true
|
||||
|
||||
<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">‣ 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>
|
||||
|
||||
@ -259,7 +287,7 @@ true
|
||||
|
||||
<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">‣ 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">‣ 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">‣ 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">‣ 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">‣ 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">‣ 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">‣ 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">‣ 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>
|
||||
|
||||
@ -339,46 +367,48 @@ 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">‣ 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">‣ 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">‣ 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">‣ 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>
|
||||
|
||||
@ -105,9 +105,6 @@
|
||||
[33X[0;0Y[10X( IsLeftAutomorphicLoop, IsAutomorphicLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsRightAutomorphicLoop, IsAutomorphicLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsMiddleAutomorphicLoop, IsAutomorphicLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsMiddleAutomorphicLoop, IsCommutative )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsCommutative )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsRightAutomorphicLoop and IsCommutative )[110X[133X
|
||||
[33X[0;0Y[10X( IsLeftAutomorphicLoop, IsRightAutomorphicLoop and
|
||||
HasAntiautomorphicInverseProperty )[110X[133X
|
||||
[33X[0;0Y[10X( IsRightAutomorphicLoop, IsLeftAutomorphicLoop and
|
||||
@ -120,9 +117,13 @@
|
||||
[33X[0;0Y[10X( IsMoufangLoop, IsAutomorphicLoop and HasLeftInverseProperty )[110X[133X
|
||||
[33X[0;0Y[10X( IsMoufangLoop, IsAutomorphicLoop and HasRightInverseProperty )[110X[133X
|
||||
[33X[0;0Y[10X( IsMoufangLoop, IsAutomorphicLoop and HasWeakInverseProperty )[110X[133X
|
||||
[33X[0;0Y[10X( IsMiddleAutomorphicLoop, IsCommutative )[110X[133X
|
||||
[33X[0;0Y[10X( IsLeftAutomorphicLoop, IsLeftBruckLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsLeftAutomorphicLoop, IsLCCLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsRightAutomorphicLoop, IsRightBruckLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsRightAutomorphicLoop, IsRCCLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsCommutative and IsMoufangLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsLeftAutomorphicLoop and IsMiddleAutomorphicLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsRightAutomorphicLoop and IsMiddleAutomorphicLoop )[110X[133X
|
||||
[33X[0;0Y[10X( IsAutomorphicLoop, IsAssociative )[110X[133X
|
||||
|
||||
|
||||
File diff suppressed because one or more lines are too long
@ -42,6 +42,9 @@
|
||||
less than 64[117X, Nova Science Publishers Inc., Commack, NY (1999), xviii+287
|
||||
pages.
|
||||
|
||||
[[20XGre14[120X] [16XGreer, M.[116X, [17XA class of loops categorically isomorphic to Bruck loops
|
||||
of odd order[117X, [18XComm. Algebra[118X, [19X42[119X, 8 (2014), 3682–3697.
|
||||
|
||||
[[20XGKN14[120X] [16XGrishkov, A., Kinyon, M. and Nagy, G. P.[116X, [17XSolvability of commutative
|
||||
automorphic loops[117X, [18XProc. Amer. Math. Soc.[118X, [19X142[119X, 9 (2014), 3029–3037.
|
||||
|
||||
@ -89,6 +92,10 @@
|
||||
[[20XSZ12[120X] [16XSlattery, M. and Zenisek, A.[116X, [17XMoufang loops of order 243[117X,
|
||||
[18XCommentationes Mathematicae Universitatis Carolinae[118X, [19X53[119X, 3 (2012), 423–428.
|
||||
|
||||
[[20XSV17[120X] [16XStuhl, I. and Vojtěchovský, P.[116X, [17XInvolutory latin quandles, Bruck
|
||||
loops and commutative automorphic loops of odd prime power order[117X, [18X[118X (2017),
|
||||
((preprint)).
|
||||
|
||||
[[20XVoj06[120X] [16XVojtěchovský, P.[116X, [17XToward the classification of Moufang loops of
|
||||
order 64[117X, [18XEuropean J. Combin.[118X, [19X27[119X, 3 (2006), 444–460.
|
||||
|
||||
|
||||
@ -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>,
|
||||
|
||||
@ -11,7 +11,7 @@
|
||||
alternative loop, left 7.4
|
||||
alternative loop, right 7.4
|
||||
antiautomorphic inverse property 7.2-5
|
||||
[2XAreEqualDiscriminators[102X 6.11-9
|
||||
[2XAreEqualDiscriminators[102X 6.11-11
|
||||
[2XAssociatedLeftBruckLoop[102X 8.1-1
|
||||
[2XAssociatedRightBruckLoop[102X 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
|
||||
[2XAutomorphicLoop[102X 9.10-1
|
||||
[2XAutomorphicLoop[102X 9.11-1
|
||||
[2XAutomorphismGroup[102X 6.11-5
|
||||
Bol loop, left 3.3
|
||||
Bol loop, left 7.4
|
||||
@ -39,7 +39,7 @@
|
||||
Cayley table, canonical 4.3-1
|
||||
[2XCayleyTable[102X 5.1-2
|
||||
[2XCayleyTableByPerms[102X 4.6-1
|
||||
[2XCCLoop[102X 9.6-3
|
||||
[2XCCLoop[102X 9.7-3
|
||||
center 2.3
|
||||
[2XCenter[102X 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
|
||||
[2XCodeLoop[102X 9.4-1
|
||||
[2XCodeLoop[102X 9.5-1
|
||||
commutant 2.3
|
||||
[2XCommutant[102X 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
|
||||
[2XConjugacyClosedLoop[102X 9.6-3
|
||||
[2XConjugacyClosedLoop[102X 9.7-3
|
||||
conjugation 6.5
|
||||
coset 6.2-6
|
||||
derived series 2.4
|
||||
@ -64,7 +64,7 @@
|
||||
[2XDerivedSubloop[102X 6.10-2
|
||||
diassociative quasigroup 7.1-4
|
||||
[2XDirectProduct[102X 4.11-1
|
||||
[2XDiscriminator[102X 6.11-8
|
||||
[2XDiscriminator[102X 6.11-10
|
||||
[2XDisplayLibraryInfo[102X 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
|
||||
[2XInnerMappingGroup[102X 6.5-3
|
||||
[2XInterestingLoop[102X 9.11-1
|
||||
[2XInterestingLoop[102X 9.12-1
|
||||
[2XIntoGroup[102X 4.10-4
|
||||
[2XIntoLoop[102X 4.10-3
|
||||
[2XIntoQuasigroup[102X 4.10-1
|
||||
@ -169,8 +169,8 @@
|
||||
[2XIsNilpotent[102X 6.9-1
|
||||
[2XIsNormal[102X 6.7-1
|
||||
[2XIsNuclearSquareLoop[102X 7.4-11
|
||||
[2XIsomorphicCopyByNormalSubloop[102X 6.11-7
|
||||
[2XIsomorphicCopyByPerm[102X 6.11-6
|
||||
[2XIsomorphicCopyByNormalSubloop[102X 6.11-9
|
||||
[2XIsomorphicCopyByPerm[102X 6.11-8
|
||||
isomorphism 2.6
|
||||
[2XIsomorphismLoops[102X 6.11-2
|
||||
[2XIsomorphismQuasigroups[102X 6.11-1
|
||||
@ -206,16 +206,17 @@
|
||||
[2XIsSubquasigroup[102X 6.2-3
|
||||
[2XIsTotallySymmetric[102X 7.3-2
|
||||
[2XIsUnipotent[102X 7.3-5
|
||||
[2XItpSmallLoop[102X 9.12-1
|
||||
[2XItpSmallLoop[102X 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
|
||||
[2XLCCLoop[102X 9.6-2
|
||||
[2XLCCLoop[102X 9.7-2
|
||||
[2XLeftBolLoop[102X 9.2-1
|
||||
[2XLeftConjugacyClosedLoop[102X 9.6-2
|
||||
[2XLeftBruckLoop[102X 9.3-1
|
||||
[2XLeftConjugacyClosedLoop[102X 9.7-2
|
||||
[2XLeftDivision[102X 5.2-1
|
||||
[2XLeftDivision[102X 5.2-1
|
||||
[2XLeftDivision[102X 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 @@
|
||||
[2XLoopByRightFolder[102X 4.7-1
|
||||
[2XLoopByRightSection[102X 4.6-3
|
||||
[2XLoopFromFile[102X 4.5-1
|
||||
[2XLoopIsomorph[102X 6.11-7
|
||||
[2XLoopMG2[102X 8.2-3
|
||||
[2XLoopsUpToIsomorphism[102X 6.11-4
|
||||
[2XLoopsUpToIsotopism[102X 6.12-2
|
||||
@ -299,7 +301,7 @@
|
||||
modification, cyclic 8.2-1
|
||||
modification, dihedral 8.2-2
|
||||
Moufang loop 7.4
|
||||
[2XMoufangLoop[102X 9.3-1
|
||||
[2XMoufangLoop[102X 9.4-1
|
||||
multiplication group 2.2
|
||||
multiplication group, left 2.2
|
||||
multiplication group, relative 6.4-2
|
||||
@ -315,7 +317,7 @@
|
||||
[2XNilpotencyClassOfLoop[102X 6.9-2
|
||||
nilpotent loop 2.4
|
||||
nilpotent loop, strongly 6.9-3
|
||||
[2XNilpotentLoop[102X 9.9-1
|
||||
[2XNilpotentLoop[102X 9.10-1
|
||||
normal closure 6.7-2
|
||||
normal subloop 6.7-1
|
||||
[2XNormalClosure[102X 6.7-2
|
||||
@ -332,7 +334,7 @@
|
||||
nucleus, right 2.3
|
||||
[2XNucleusOfLoop[102X 6.6-2
|
||||
[2XNucleusOfQuasigroup[102X 6.6-2
|
||||
octonion loop 9.3-1
|
||||
octonion loop 9.4-1
|
||||
[2XOne[102X 5.1-3
|
||||
[2XOneLoopTableInGroup[102X 8.4-3
|
||||
[2XOneLoopWithMltGroup[102X 8.4-6
|
||||
@ -342,8 +344,8 @@
|
||||
[2XOppositeLoop[102X 4.12-1
|
||||
[2XOppositeQuasigroup[102X 4.12-1
|
||||
Osborn loop 7.6-4
|
||||
Paige loop 9.8
|
||||
[2XPaigeLoop[102X 9.8-1
|
||||
Paige loop 9.9
|
||||
[2XPaigeLoop[102X 9.9-1
|
||||
[2XParent[102X 6.1-1
|
||||
[2XPosInParent[102X 6.1-3
|
||||
[2XPosition[102X 6.1-2
|
||||
@ -373,18 +375,20 @@
|
||||
[2XQuasigroupByRightFolder[102X 4.7-1
|
||||
[2XQuasigroupByRightSection[102X 4.6-3
|
||||
[2XQuasigroupFromFile[102X 4.5-1
|
||||
[2XQuasigroupIsomorph[102X 6.11-6
|
||||
[2XQuasigroupsUpToIsomorphism[102X 6.11-3
|
||||
[2XRandomLoop[102X 4.9-1
|
||||
[2XRandomNilpotentLoop[102X 4.9-2
|
||||
[2XRandomQuasigroup[102X 4.9-1
|
||||
RC loop 7.4
|
||||
[2XRCCLoop[102X 9.6-1
|
||||
[2XRCCLoop[102X 9.7-1
|
||||
[2XRelativeLeftMultiplicationGroup[102X 6.4-2
|
||||
[2XRelativeMultiplicationGroup[102X 6.4-2
|
||||
[2XRelativeRightMultiplicationGroup[102X 6.4-2
|
||||
[2XRightBolLoop[102X 9.2-2
|
||||
[2XRightBolLoopByExactGroupFactorization[102X 8.1-3
|
||||
[2XRightConjugacyClosedLoop[102X 9.6-1
|
||||
[2XRightBruckLoop[102X 9.3-2
|
||||
[2XRightConjugacyClosedLoop[102X 9.7-1
|
||||
[2XRightCosets[102X 6.2-6
|
||||
[2XRightDivision[102X 5.2-1
|
||||
[2XRightDivision[102X 5.2-1
|
||||
@ -400,7 +404,7 @@
|
||||
[2XRightTransversal[102X 6.2-7
|
||||
section, left 2.2
|
||||
section, right 2.2
|
||||
sedenion loop 9.11
|
||||
sedenion loop 9.12
|
||||
semisymmetric quasigroup 7.3-1
|
||||
[2XSetLoopElmName[102X 3.4-1
|
||||
[2XSetQuasigroupElmName[102X 3.4-1
|
||||
@ -408,12 +412,12 @@
|
||||
simple loop 6.7-3
|
||||
[2XSize[102X 5.1-4
|
||||
[2XSmallGeneratingSet[102X 5.5-3
|
||||
[2XSmallLoop[102X 9.7-1
|
||||
[2XSmallLoop[102X 9.8-1
|
||||
solvability class 2.4
|
||||
solvable loop 2.4
|
||||
Steiner loop 7.8-2
|
||||
Steiner quasigroup 7.3-4
|
||||
[2XSteinerLoop[102X 9.5-1
|
||||
[2XSteinerLoop[102X 9.6-1
|
||||
strongly nilpotent loop 6.9-3
|
||||
subloop 2.3
|
||||
[2XSubloop[102X 6.2-2
|
||||
|
||||
@ -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
161
doc/loops.bbl
@ -1,161 +0,0 @@
|
||||
\begin{thebibliography}{DBGV12}
|
||||
|
||||
\bibitem[Art59]{Ar}
|
||||
R.~Artzy.
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||||
\newblock On automorphic-inverse properties in loops.
|
||||
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||||
|
||||
\bibitem[Art15]{Artic}
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||||
K.~Artic.
|
||||
\newblock {\em On conjugacy closed loops and conjugacy closed loop folders}.
|
||||
\newblock PhD thesis, RWTH Aachen University, 2015.
|
||||
|
||||
\bibitem[BP56]{BrPa}
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||||
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.
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||||
|
||||
\bibitem[Bru58]{Br}
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||||
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\newblock Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft
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\bibitem[CD05]{CsDr}
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P.~Cs{\"o}rg{\H o} and A.~Dr{\a'a}pal.
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\newblock Left conjugacy closed loops of nilpotency class two.
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||||
\newblock {\em Results Math.}, 47(3-4):242{\textendash}265, 2005.
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||||
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||||
\bibitem[CR99]{CoRo}
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\newblock {\em Triple systems}.
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\newblock Commutative automorphic loops of order {$p^3$}.
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||||
\newblock {\em J. Algebra Appl.}, 11(5):1250100, 15, 2012.
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\newblock Cyclic and dihedral constructions of even order.
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||||
\newblock Moufang loops that share associator and three quarters of their
|
||||
multiplication tables.
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||||
\newblock {\em Rocky Mountain J. Math.}, 36(2):425{\textendash}455, 2006.
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\newblock Enumeration of nilpotent loops via cohomology.
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\bibitem[Fen69]{Fe}
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\newblock Extra loops. {II}. {O}n loops with identities of {B}ol-{M}oufang
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||||
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\newblock Solvability of commutative automorphic loops.
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||||
\newblock {\em Proc. Amer. Math. Soc.}, 142(9):3029{\textendash}3037, 2014.
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\bibitem[GMR99]{Go}
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\newblock {\em The {M}oufang loops of order less than 64}.
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\newblock Searching for small simple automorphic loops.
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\newblock {\em LMS J. Comput. Math.}, 14:200{\textendash}213, 2011.
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\newblock Nilpotency in automorphic loops of prime power order.
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\newblock Generating uniformly distributed random {L}atin squares.
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M.~K. Kinyon, K.~Kunen, and J.~D. Phillips.
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\newblock Every diassociative {$A$}-loop is {M}oufang.
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||||
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\newblock The structure of automorphic loops.
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||||
\newblock Bol loops and bruck loops of order $pq$.
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||||
\newblock 2015.
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||||
\newblock preprint.
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||||
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||||
\bibitem[Kun00]{Kun}
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||||
\bibitem[Lie87]{Li}
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\newblock The classification of finite simple {M}oufang loops.
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||||
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||||
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||||
\bibitem[Moo]{Mo}
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||||
G.~E. Moorhouse.
|
||||
\newblock Bol loops of small order.
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||||
\newblock http://www.uwyo.edu/moorhouse/pub/bol/.
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||||
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||||
\bibitem[NV03]{NaVo2003}
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M.~Slattery and A.~Zenisek.
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\newblock Moufang loops of order 243.
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P.~Vojt{\v e}chovsk{\a'y}.
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\end{thebibliography}
|
||||
@ -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 $64$},
|
||||
|
||||
@ -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)
|
||||
@ -1,35 +0,0 @@
|
||||
\backcite {Br}{{8}{2}{chapter.2}}
|
||||
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|
||||
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|
||||
\backcite {Vo}{{35}{6.11.8}{subsection.6.11.8}}
|
||||
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|
||||
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|
||||
\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}}
|
||||
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|
||||
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|
||||
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|
||||
\backcite {VoQRS}{{43}{7.7}{section.7.7}}
|
||||
\backcite {JeKiVo}{{43}{7.7}{section.7.7}}
|
||||
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|
||||
\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
427
doc/loops.idx
@ -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}
|
||||
@ -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
485
doc/loops.ind
@ -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
856
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|
||||
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
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||||
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croatian, czech, danish, dutch, esperanto, estonian, farsi, finnish, french, ga
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lician, german, german-x-2013-05-26, greek, gujarati, hindi, hungarian, iceland
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lithuanian, malayalam, marathi, mongolian, mongolianlmc, monogreek, ngerman, n
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Appendix A.
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@ -1,83 +0,0 @@
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\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 1
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\BOOKMARK [1][-]{section.1.1}{License}{chapter.1}% 2
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\BOOKMARK [1][-]{section.1.2}{Installation}{chapter.1}% 3
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|
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\BOOKMARK [0][-]{chapter.2}{Mathematical Background}{}% 9
|
||||
\BOOKMARK [1][-]{section.2.1}{Quasigroups and Loops}{chapter.2}% 10
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||||
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\BOOKMARK [1][-]{section.2.3}{Subquasigroups and Subloops}{chapter.2}% 12
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\BOOKMARK [1][-]{section.2.4}{Nilpotence and Solvability}{chapter.2}% 13
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\BOOKMARK [1][-]{section.2.5}{Associators and Commutators}{chapter.2}% 14
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\BOOKMARK [1][-]{section.2.6}{Homomorphism and Homotopisms}{chapter.2}% 15
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\BOOKMARK [0][-]{chapter.3}{How the Package Works}{}% 16
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||||
\BOOKMARK [1][-]{section.3.1}{Representing Quasigroups}{chapter.3}% 17
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||||
\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
|
||||
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|
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\BOOKMARK [1][-]{section.4.1}{About Cayley Tables}{chapter.4}% 22
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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||||
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|
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|
||||
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|
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|
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|
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|
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|
||||
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||||
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|
||||
\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
|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
\BOOKMARK [1][-]{section.9.11}{Interesting Loops}{chapter.9}% 78
|
||||
\BOOKMARK [1][-]{section.9.12}{Libraries of Loops Up To Isotopism}{chapter.9}% 79
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||||
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|
||||
\BOOKMARK [0][-]{appendix*.3}{References}{}% 82
|
||||
\BOOKMARK [0][-]{section*.4}{Index}{}% 83
|
||||
246
doc/loops.pnr
246
doc/loops.pnr
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3930
doc/loops.tex
3930
doc/loops.tex
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241
doc/loops.toc
241
doc/loops.toc
@ -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}
|
||||
@ -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>Gábor 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>©right; 2016 Gábor P. Nagy and Petr Vojtěchovský.
|
||||
<Copyright>©right; 2017 Gábor P. Nagy and Petr Vojtěchovský.
|
||||
</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, Aleš Drápal, Steve Flammia, Kenneth W. Johnson, Michael K. Kinyon, Alexander Konovalov, Frank Lübeck 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, 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/>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/>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.
|
||||
|
||||
@ -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>. Csörgő and Drápal <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>
|
||||
|
||||
|
||||
@ -1,559 +0,0 @@
|
||||
<?xml version="1.0" encoding="UTF-8"?>
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}
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}
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|
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}
|
||||
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|
||||
}
|
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|
||||
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|
||||
}
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||||
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}
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}
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}
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}
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||||
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@ -258,134 +258,138 @@ X", "6.7", [ 6, 7, 0 ], 320, 31, "normal subloops and simple loops",
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||||
, "6.11", [ 6, 11, 0 ], 444, 33, "isomorphisms and automorphisms",
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||||
33X\033[101X", "7", [ 7, 0, 0 ], 1, 37,
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[
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"\033[1X\033[33X\033[0;-2YAssociativity, Commutativity and Generalizations\\
|
||||
033[133X\033[101X", "7.1", [ 7, 1, 0 ], 16, 36,
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||||
033[133X\033[101X", "7.1", [ 7, 1, 0 ], 16, 37,
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"associativity commutativity and generalizations", "X7960E3FB7A7F0F00" ]
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, [ "\033[1X\033[33X\033[0;-2YInverse Propeties\033[133X\033[101X",
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||||
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[
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||||
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||||
, "7.3", [ 7, 3, 0 ], 102, 38, "some properties of quasigroups",
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||||
, "7.3", [ 7, 3, 0 ], 102, 39, "some properties of quasigroups",
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||||
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[
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||||
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||||
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"7.3-7", [ 7, 3, 7 ], 160, 39, "isentropic and ismedial",
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"7.3-7", [ 7, 3, 7 ], 160, 40, "isentropic and ismedial",
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"X7F23D4D97A38D223" ],
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[ "\033[1X\033[33X\033[0;-2YLoops of Bol Moufang Type\033[133X\033[101X",
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||||
"7.4", [ 7, 4, 0 ], 170, 39, "loops of bol moufang type",
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||||
"7.4", [ 7, 4, 0 ], 170, 40, "loops of bol moufang type",
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"X780D907986EBA6C7" ],
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[ "\033[1X\033[33X\033[0;-2YPower Alternative Loops\033[133X\033[101X",
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||||
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||||
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|
||||
[
|
||||
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||||
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||||
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||||
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||||
"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",
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[ 7, 7, 0 ], 384, 43, "automorphic loops", "X793B22EA8643C667" ],
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[ 7, 7, 0 ], 384, 44, "automorphic loops", "X793B22EA8643C667" ],
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"7.8", [ 7, 8, 0 ], 451, 44, "additonal varieties of loops",
|
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|
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[
|
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"\033[1X\033[33X\033[0;-2YIsLeftBruckLoop and IsLeftKLoop\033[133X\033[101X\
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", "7.8-3", [ 7, 8, 3 ], 470, 44, "isleftbruckloop and isleftkloop",
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|
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[
|
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"\033[1X\033[33X\033[0;-2YIsRightBruckLoop and IsRightKLoop\033[133X\033[10\
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|
||||
"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" ] ]
|
||||
);
|
||||
|
||||
@ -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
|
||||
|
||||
@ -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",
|
||||
|
||||
@ -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 );
|
||||
|
||||
@ -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" );
|
||||
|
||||
211
gap/examples.gi
211
gap/examples.gi
@ -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 );
|
||||
# coordinates determining the cocycle
|
||||
dim := Length( LOOPS_automorphic_bases[ pos_n ][ factor_id ] );
|
||||
coords := LOOPS_automorphic_coordinates[ pos_n ][ m ];
|
||||
coords := coords{[2..Length(coords)]}; # remove the character that determines factor id
|
||||
coords := LOOPS_ConvertBase( coords, 91, 3, dim );
|
||||
coords := List( coords, LOOPS_CharToDigit );
|
||||
# basis
|
||||
basis := List( LOOPS_automorphic_bases[ pos_n ][ factor_id ],
|
||||
i -> LOOPS_automorphic_cocycles[ pos_n ][ i ]
|
||||
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
|
||||
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, Length( basis ) );
|
||||
coords := List( coords, LOOPS_CharToDigit );
|
||||
# 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;
|
||||
|
||||
# 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
|
||||
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
|
||||
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);
|
||||
|
||||
@ -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
65
gap/iso.gi
@ -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);
|
||||
|
||||
|
||||
@ -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);
|
||||
|
||||
@ -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);
|
||||
|
||||
@ -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 ), ">");
|
||||
PrintMe("alternative", L );
|
||||
else
|
||||
Print( "<left alternative loop of order ", Size( L ), ">");
|
||||
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 ), ">" );
|
||||
|
||||
@ -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
|
||||
|
||||
|
||||
@ -2,7 +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)
|
||||
|
||||
@ -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
54
tst/lib.tst
@ -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 );
|
||||
|
||||
@ -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" ) );
|
||||
|
||||
Loading…
Reference in New Issue
Block a user