If you do not see the subfolder pkg/loops in the main directory of ⪆ then download the LOOPS package from the distribution website http://www.math.du.edu/loops and unpack the downloaded file into the pkg subfolder.

The package LOOPS can then be loaded to ⪆ anytime by calling LoadPackage("loops");.

If you wish to load LOOPS automatically while starting ⪆, start ⪆, execute the following commands, gap> pref := UserPreference( "PackagesToLoad " );; gap> Add( pref, "loops" );; gap> SetUserPreference( "PackagesToLoad", pref );; gap> WriteGapIniFile();; quit ⪆ and restart it.

The preferred format of the documentation is html with MathJax turned on.

All formats of the documentation can be found in the doc folder of LOOPS. You can also download the documentation from the LOOPS distribution website.

The inline ⪆ help is available upon installing LOOPS and can be accessed in the usual way, i.e., upon typing ?command, ⪆ displays the section of the LOOPS manual containing information about command.

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.

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.

In ⪆, functions of the type AsSomething(X) convert the domain X into Something, if possible, without changing the underlying domain X. For example, if X is declared as magma but is associative and has neutral element and inverses, AsGroup(X) returns the corresponding group with the underlying domain X.

We have opted for a more general kind of conversions in LOOPS (starting with version 2.1.0), using functions of the type IntoSomething(X). The two main features that distinguish IntoSomething from AsSomething are: The function IntoSomething(X) does not necessarily return the same domain as X. The reason is that X can be a group, for instance, defined on one of many possible domains, while IntoLoop(X) must result in a loop, and hence be defined on a subset of some interval 1, \dots, n (see Section ). In some special situations, the function IntoSomething(X) allows to convert X into Something even though X does not have all the properties of Something. For instance, every quasigroup is isotopic to a loop, so it makes sense to allow the conversion IntoLoop(Q) even if the quasigroup Q does not posses a neutral element. Details of all conversions in LOOPS can be found in Section .

Only loops contained in the libraries of LOOPS are named. For instance, the loop obtained via MoufangLoop(32,4), the 4th Moufang loop of order 32, is named "Moufang loop 32/4'' and is shown as <Moufang loop 32/4>.

A generic quasigroup of order n is displayed as <quasigroup of order n>. Similarly, a loop of order n appears as <loop of order n>.

The displayed information of a generic loop is enhanced if more information about the loop becomes available. For instance, when it is established that a loop of order 12 has the left alternative property, the loop will be shown as <left alternative loop of order 12> until a stronger property is obtained. Which property is diplayed is governed by the filters built into LOOPS (see Appendix ). SetQuasigroupElmName and SetLoopElmName The above functions change the names of elements of a quasigroup (resp. loop) Q to name. By default, elements of a quasigroup appear as qi and elements of a loop appear as li in both display modes, where i is a positive integer. The neutral element of a loop is always indexed by 1.

For quasigroups and loops in the Print mode, we display the multiplication table (if it is known), otherwise we display the elements.

In the following example, L is a loop with two elements. gap> L; <loop of order 2> gap> Print( L ); <loop with multiplication table [ [ 1, 2 ], [ 2, 1 ] ]> gap> Elements( L ); [ l1, l2 ] gap> SetLoopElmName( L, "loop_element" );; Elements( L ); [ loop_element1, loop_element2 ]

A Cayley table is a quasigroup tablequasigroup table if it is a latin squarelatin square, i.e., if every entry x_i appears in every column and every row exactly once.

An unfortunate feature of multiplication tables in practice is that they are often not bordered, that is, it is up to the reader to figure out what is meant. Throughout this manual and in LOOPS, we therefore make the following assumption: All distinct entries in a quasigroup table must be positive integers, say x_1 < x_2 < \cdots < x_n, and if no border is specified, we assume that the table is bordered by x_1, \dots, x_n, in this order. Note that we do not assume that the distinct entries x_1, \dots, x_n form the interval 1, \dots, n. The significance of this observation will become clear in Chapter .

Finally, we say that a quasigroup table is a loop tableloop table if the first row and the first column are the same, and if the entries in the first row are ordered in an ascending fashion.

Remark:The package GUAVA also contains operations dealing with latin squares. In particular, IsLatinSquare is declared in GUAVA.

Instead of writing a separate algorithm for each common format, our algorithm relies on the user to provide a bit of information about the input file. Here is an outline of the algorithm, with file named filename and a string del as input (in essence, the characters of del will be ignored while reading the file): read the entire content of filename into a string s, replace all end-of-line characters in s by spaces, replace by spaces all characters of s that appear in del, split s into maximal substrings without spaces, called chunks here, let n be the number of distinct chunks, if the number of chunks is not n^2, report error, construct the multiplication table by assigning numerical values 1, \dots, n to chunks, depending on their position among distinct chunks.

The following examples clarify the algorithm and document its versatility. All examples are of the form F+D\Longrightarrow T, meaning that an input file containing F together with the deletion string D produce multiplication table T.

Example: Data does not have to be arranged into an array of any kind. \begin{array}{cccc} 0&1&2&1\\ 2&0&2& \\ 0&1& & \end{array}\quad + \quad "" \quad \Longrightarrow\quad \begin{array}{ccc} 1&2&3\\ 2&3&1\\ 3&1&2 \end{array}

Example: Chunks can be any strings. \begin{array}{cc} {\rm red}&{\rm green}\\ {\rm green}&{\rm red}\\ \end{array}\quad + \quad "" \quad \Longrightarrow\quad \begin{array}{cc} 1& 2\\ 2& 1 \end{array}

Example: A typical table produced by &GAP; is easily parsed by deleting brackets and commas. [ [0, 1], [1, 0] ] \quad + \quad "[,]" \quad \Longrightarrow\quad \begin{array}{cc} 1& 2\\ 2& 1 \end{array}

Example: A typical &TeX; table with rows separated by lines is also easily converted. Note that we have to use \backslash\backslash to ensure that every occurrence of \backslash is deleted, since \backslash\backslash represents the character \backslash in &GAP; \begin{array}{lll} x\&& y\&&\ z\backslash\backslash\cr y\&& z\&&\ x\backslash\backslash\cr z\&& x\&&\ y \end{array} \quad + \quad "\backslash\backslash\&" \quad \Longrightarrow\quad \begin{array}{ccc} 1&2&3\cr 2&3&1\cr 3&1&2 \end{array} QuasigroupFromFile and LoopFromFile The quasigroup (resp. loop) whose multiplication table data is in file filename, ignoring the characters contained in the string del.

We call the triple (G,H,T) a folderright quasigroup (or loop) folder. QuasigroupByRightFolder and LoopByRightFolder The quasigroup (resp. loop) from the right folder (G, H, T).

Remark: We do not support the dual operations for left sections since, by default, actions in &GAP; act on the right.

Here is a simple example in which T is actually the right section of the resulting loop. gap> T := [ (), (1,2)(3,4,5), (1,3,5)(2,4), (1,4,3)(2,5), (1,5,4)(2,3) ];; gap> G := Group( T );; H := Stabilizer( G, 1 );; gap> LoopByRightFolder( G, H, T ); <loop of order 5>

A map \theta:F\times F\to K is a cocyclecocycle if \theta(1,x) = \theta(x,1) = 1 for every x\in F.

The following theorem holds for loops Q, F and an abelian group K: Q is a nuclear extension of K by F if and only if there is a cocycle \theta:F\times F\to K and a homomorphism \varphi:F\to{\rm Aut}(Q) such that K\times F with multiplication (a,x)(b,y) = (a\varphi_x(b)\theta(x,y),xy) is isomorphic to Q. The data necessary to construct Q as a nuclear extension of the subloop K by Q/K, namely [K, F, \varphi, \theta] as above. Note that K must be a commutative subloop of the nucleus of Q. If n=|F| and m=|K|, the cocycle \theta is returned as an n\times n array with entries in \{1,\dots,m\}, and the homomorphism \varphi is returned as a list of length n of permutations of \{1,\dots,m\}. The extension of an abelian group K by a loop F, using action f and cocycle t. The arguments must be formatted as the output of NuclearExtension. gap> F := IntoLoop( Group( (1,2) ) ); <loop of order 2> gap> K := DirectProduct( F, F );; gap> phi := [ (), (2,3) ];; gap> theta := [ [ 1, 1 ], [ 1, 3 ] ];; gap> LoopByExtension( K, F, phi, theta ); <loop of order 8> gap> IsAssociative( last ); false

In , Jacobson and Matthews defined a random walk on the space of latin squares and so-called improper latin squares that visits every latin square with the same probability. The diameter of the space is no more than 4(n-1)^3 in the sense that no more than 4(n-1)^3 properly chosen steps are needed to travel from one latin square of order n to another.

The Jacobson-Matthews algorithm can be used to generate random quasigroups as follows: (i) select any latin square of order n, for instance the canonical multiplication table of the cyclic group of order n, (ii) perform sufficiently many steps of the random walk, stopping at a proper or improper latin square, (iii) if necessary, perform a few more steps to end up with a proper latin square. Upon normalizing the resulting latin square, we obtain a random loop of order n.

By the above result, it suffices to use about n^3 steps to arrive at any latin square of order n from the initial latin square. In fact, a smaller number of steps is probably sufficient. RandomQuasigroup and RandomLoop A random quasigroup (resp. loop) of order n using the Jacobson-Matthews algorithm. If the optional argument iter is omitted, n{}^3 steps are used. Otherwise iter steps are used. If iter is small, the Cayley table of the returned quasigroup (resp. loop) will be close to the canonical Cayley table of the cyclic group of order n. A random nilpotent looploop as follows (see Section for more information on nilpotency): lst must be a list of positive integers and/or finite abelian groups. If lst=[a1] and a1 is an integer, a random abelian group of order a1 is returned, else a1 is an abelian group and AsLoop(a1) is returned. If lst= [a1,...,am], a random central extension of RandomNilpotentLoop([a1]) by RandomNilpotentLoop([a2,...,am]) is returned. To determine the nilpotency class c of the resulting loop, assume that lst has length at least 2, contains only integers bigger than 1, and let m be the last entry of lst. If m>2 then c is equal to Length(lst), else c is equal to Length(lst)-1.

Remark: Up to version 2.0.0 of LOOPS, we supported AsQuasigroup, AsLoop and AsGroup in place of IntoQuasigroup, IntoLoop and IntoGroup, respectively. We have changed the terminology starting with version 2.1.0 in order to comply with &GAP; naming rules for AsSomething, as explained in Chapter . Finally, the method AsGroup is a core method of &GAP; that returns an fp group if its argument is an associative loop. If M is a declared magma that happens to be a quasigroup, the corresponding quasigroup is returned. If M is already declared as a quasigroup, M is returned. An isomorphic copy of the principal isotope (M,\circ) via the transposition (1,f\cdotg). An isomorphic copy is returned rather than (M,\circ) because in LOOPS all loops have to have neutral element labeled as 1. Given a quasigroup M and two of its elements f, g, the principal loop isotope x\circ y = R_g^{-1}(x)\cdot L_f^{-1}(y) turns (M,\circ) into a loop with neutral element f\cdot g (see Section ). If M is a declared magma that happens to be a quasigroup (but not necessarily a loop!), a loop is returned as follows: If M is already declared as a loop, M is returned. Else, if M possesses a neutral element e and if f is the first element of M, then an isomorphic copy of M via the transposition (e,f) is returned. If M does not posses a neutral element, PrincipalLoopIsotope(M, M.1, M.1) is returned.

Remark: One could obtain a loop from a declared magma M in yet another way, by normalizing the Cayley table of M. The three approaches can result in nonisomorphic loops in general. If M is a declared magma that happens to be a group, the corresponding group is returned as follows: If M is already declared as a group, M is returned, else RightMultiplicationGroup(IntoLoop(M)) is returned, which is a permutation group isomorphic to M.

Remark:If you want to know if a quasigroup Q has a neutral element, you can find out with the standard function for magmas MultiplicativeNeutralElement(Q). The size of a quasigroup Q. The exponent of a power associative loop Q. (The method does not test if Q is power associative.) When Q is a power associative looplooppower associative loop, that is, the powers of elements are well-defined in Q, then the exponentexponent of Q is the smallest positive integer divisible by the orders of all elements of Q.

Two elements x, y of the same quasigroup are multiplied by x*y in &GAP;. Since multiplication of at least three elements is ambiguous in the nonassociative case, we parenthesize elements by default from left to right, i.e., x*y*z means ((x*y)*z). Of course, one can specify the order of multiplications by providing parentheses. LeftDivision and RightDivision The left division x\backslashy (resp. the right division x/y) of two elements x, y of the same quasigroup.
The list of elements obtained by performing the specified arithmetical operation elementwise using a list S of elements and an element x.

Remark: We support / in place of RightDivision. But we do not support \backslash in place of LeftDivision. LeftDivisionCayleyTable and RightDivisionCayleyTable The Cayley table of the respective arithmetic operation of a quasigroup Q.

Let x be an element of a loop Q with neutral element 1. Then the left inverseinverse x^\lambda of x is the unique element of Q satisfying x^\lambda x=1. Similarly, the right inverseinverse x^\rho satisfies xx^\rho=1. If x^\lambda=x^\rho, we call x^{-1}=x^\lambda=x^\rho the inverseinverse of x. LeftInverse, RightInverse and Inverse The left inverse, right inverse and inverse, respectively, of the quasigroup element x. gap> CayleyTable( Q ); [ [ 1, 2, 3, 4, 5 ], [ 2, 1, 4, 5, 3 ], [ 3, 4, 5, 1, 2 ], [ 4, 5, 2, 3, 1 ], [ 5, 3, 1, 2, 4 ] ] gap> elms := Elements( Q ); gap> [ l1, l2, l3, l4, l5 ]; gap> [ LeftInverse( elms[3] ), RightInverse( elms[3] ), Inverse( elms[3] ) ]; [ l5, l4, fail ]

As usual in &GAP;, one can refer to the ith generator of a quasigroup Q by Q.i. Note that while it is often the case that Q.i = Elements(Q)[i], it is not necessarily so. A generating set \{q_0, \dots, q_m\} of Q such that Q_0=\emptyset, Q_m=Q, Q_i=\langle q_1, \dots, q_i \rangle, and q_{i+1} is the least element of Q\setminus Q_i. A small generating set \{q_0, \dots, q_m\} of Q obtained as follows: q_0 is the least element for which \langle q_0\rangle is largest possible, q_1$ is the least element for which \langle q_0,q_1 is largest possible, and so on.

See Appendix for implications implemented in LOOPS among various inverse properties.

Left power alternative loops are left alternative and have the left inverse property. Left Bol loops and LC loops are left power alternative. IsLeftPowerAlternative, IsRightPowerAlternative and IsPowerAlternative true if Q is a left power alternative loop, resp. a right power alternative loop, resp. a power alternative loop.

The equivalence LCC + RCC = CC is built into LOOPS. true if Q is a left conjugacy closed loop. true if Q is a right conjugacy closed loop. true if Q is a conjugacy closed loop. true if Q is an Osborn loop. A loop is OsbornOsborn looploop if it satisfies x(yz\cdot x)=(x^\lambda\backslash y)(zx). Both Moufang loops and CC loops are Osborn.

A loop Q is an automorphic loopautomorphic looploop if all inner mappings of Q are automorphisms of Q.

Automorphic loops are also known as A loops, and similar terminology exists for left, right and middle automorphic loops.

The following results hold for automorphic loops: automorphic loops are power associative in an automorphic loop Q we have {\rm Nuc}(Q) = {\rm Nuc}_{\lambda}(Q) = {\rm Nuc}_{\rho}(Q)\le {\rm Nuc}_{\mu}(Q) and all nuclei are normal a loop that is left automorphic and right automorphic satisfies the anti-automorphic inverse property and is automorphic diassociative automorphic loops are Moufang automorphic loops of odd order are solvable finite commutative automorphic loops are solvable commutative automorphic loops of order p, 2p, 4p, p^2, 2p^2, 4p^2 (p an odd prime) are abelian groups commutative automorphic loops of odd prime power order are centrally nilpotent for any prime p, there are 7 commutative automorphic loops of order p^3 up to isomorphism See the built-in filters and the survey for additional properties of automorphic loops. true if Q is a left automorphic loop. true if Q is a middle automorphic loop. true if Q is a right automorphic loop. true if Q is an automorphic loop.

Remark: Be careful not to confuse IsALoop and IsLoop.

The following operations search for loops whose multiplication groups are contained within a specified transitive permutation group G. In all these operations, one can speed up the search by increasing the optional argument depth, the price being a much higher memory consumption. The argument depth is optimally chosen if in the permutation group G there are not many permutations fixing depth elements. It is safe to omit the argument or set it equal to 2.

The optional argument infolevel determines the amount of information displayed during the search. With infolevel=0, no information is provided. With infolevel=1, you get some information on timing and hits. With infolevel=2, the results are printed as well. All Cayley tables of loops whose multiplication group is contained in the transitive permutation group G. All Cayley tables of nonassociative loops whose multiplication group is contained in the transitive permutation group G. A Cayley table of a loop whose multiplication group is contained in the transitive permutation group G. A Cayley table of a nonassociative loop whose multiplication group is contained in the transitive permutation group G. A list of all loops (given as sections) whose multiplication group is equal to the transitive permutation group G. One loop (given as a section) whose multiplication group is equal to the transitive permutation group G. gap> g:=PGL(3,3); Group([ (6,7)(8,11)(9,13)(10,12), (1,2,5,7,13,3,8,6,10,9,12,4,11) ]) gap> a:=AllLoopTablesInGroup(g,3,0);; Size(a); 56 gap> a:=AllLoopsWithMltGroup(g,3,0);; Size(a); 52

In most cases, the array LOOPS_my_library_data consists of three lists: LOOPS_my_library_data[1] is a list of orders for which there is at least one loop in the library, LOOPS_my_library_data[2][k] is the number of loops of order LOOPS_my_library_data[1][k] in the library, LOOPS_my_library_data[3][k][s] contains data necessary to produce the sth loop of order LOOPS_my_library_data[1][k] in the library. The format of LOOPS_my_library_data[3] depends heavily on the particular library and is not standardized in any way. The data is often coded to save space. The mth loop of order n from the library named libname. This is a template function that retrieves the mth loop of order n from the library named my library.

For example, the mth left Bol loop of order n is obtained via LeftBolLoop(n,m) or via LibraryLoop("left Bol",n,m). Brief information about the loops contained in the library named libname.

We are now going to describe the individual libraries.

The classification of left Bol loops of order 16 was first accomplished by Moorhouse . Our library was generated independently and it agrees with Moorhouse's results. The left Bol loops of order pq were classified in . The mth left Bol loop of order n in the library. The mth right Bol loop of order n in the library.

Remark: Only left Bol loops are stored in the library. Right Bol loops are retrieved by calling Opposite on left Bol loops.

For n\le 63, our catalog numbers coincide with those of Goodaire et al. . The classification of Moufang loops of order 64 and 81 was carried out in . The classification of Moufang loops of order 243 was carried out by Slattery and Zenisek .

The extent of the library is summarized below: \begin{array}{r|rrrrrrrrrrrrrrrrrr} order&12&16&20&24&28&32&36&40&42&44&48&52&54&56&60&64&81&243\cr loops&1 &5 &1 &5 &1 &71&4 &5 &1 &1 &51&1 &2 &4 &5 &4262& 5 &72 \end{array}

The octonion loopoctonion looploop of order 16 (i.e., the multiplication loop of the basis elements in the 8-dimensional standard real octonion algebra) can be obtained as MoufangLoop(16,3).

Our labeling of Steiner loops of order 16 coincides with the labeling of Steiner triple systems of order 15 in . The mth Steiner loop of order n in the library.

See for more on enumeration of nilpotent loops. For instance, there are 2623755 nilpotent loops of order 12, and 123794003928541545927226368 nilpotent loops of order 22. The mth nilpotent loop of order n in the library.

Note that loops up to isotopism form a subset of the corresponding library of loops up to isomorphism. For instance, the above example shows that the 14th small loop of order 6 up to isotopism is in fact the 42nd small loop of order 6 up to isomorphism.