  
  [1X3 [33X[0;0YHow the Package Works[133X[101X
  
  [33X[0;0YThe package consists of three complementary components:[133X
  
  [30X    [33X[0;6Ythe  core  algorithms for quasigroup theoretical notions (see Chapters
        [14X4[114X, [14X5[114X, [14X6[114X and [14X7[114X),[133X
  
  [30X    [33X[0;6Yalgorithms  for  specific varieties of loops, mostly for Moufang loops
        (see Chapter [14X8[114X),[133X
  
  [30X    [33X[0;6Ythe library of small loops (see Chapter [14X9[114X).[133X
  
  [33X[0;0YAlthough  we  do  not explain the algorithms in detail here, we describe the
  general  philosophy  so  that  users  can  anticipate  the  capabilities and
  behavior of [5XLOOPS[105X.[133X
  
  
  [1X3.1 [33X[0;0YRepresenting Quasigroups[133X[101X
  
  [33X[0;0YSince  permutation  representation  in  the  usual  sense  is impossible for
  nonassociative   structures,   and   since   the  theory  of  nonassociative
  presentations  is  not well understood, we resorted to multiplication tables
  to  represent  quasigroups  in  [5XGAP[105X.  (In  order  to  save storage space, we
  sometimes use one multiplication table to represent several quasigroups, for
  instance  when  a  quasigroup  is a subquasigroup of another quasigroup. See
  Section [14X4.1[114X for more details.)[133X
  
  [33X[0;0YConsequently, the package is intended primarily for quasigroups and loops of
  small order, say up to 1000.[133X
  
  [33X[0;0YThe  [5XGAP[105X  categories  [10XIsQuasigroupElement[110X,  [10XIsLoopElement[110X,  [10XIsQuasigroup[110X and
  [10XIsLoop[110X are declared in [5XLOOPS[105X as follows:[133X
  
  DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
  DeclareRepresentation( "IsQuasigroupElmRep",
      IsPositionalObjectRep and IsMultiplicativeElement, [1] );
  DeclareCategory( "IsLoopElement",
      IsQuasigroupElement and IsMultiplicativeElementWithInverse );
  DeclareRepresentation( "IsLoopElmRep",
      IsPositionalObjectRep and IsMultiplicativeElementWithInverse, [1] );
  ## latin (auxiliary category for GAP to tell apart IsMagma and IsQuasigroup)
  DeclareCategory( "IsLatin", IsObject );
  DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
  DeclareCategory( "IsLoop", IsQuasigroup and
      IsMultiplicativeElementWithInverseCollection);
  
  
  [1X3.2 [33X[0;0YConversions between magmas, quasigroups, loops and groups[133X[101X
  
  [33X[0;0YWhether  an  object  is  considered  a magma, quasigroup, loop or group is a
  matter  of  declaration  in  [5XLOOPS[105X. Loops are automatically quasigroups, and
  both  groups  and  quasigroups  are  automatically  magmas. All standard [5XGAP[105X
  commands for magmas are therefore available for quasigroups and loops.[133X
  
  [33X[0;0YIn  [5XGAP[105X,  functions  of  the  type  [10XAsSomething([3XX[103X[10X)[110X convert the domain [3XX[103X into
  [10XSomething[110X,  if  possible,  without  changing  the  underlying  domain [3XX[103X. For
  example,  if  [3XX[103X  is  declared  as  magma  but is associative and has neutral
  element  and  inverses,  [10XAsGroup([3XX[103X[10X)[110X returns the corresponding group with the
  underlying domain [3XX[103X.[133X
  
  [33X[0;0YWe have opted for a more general kind of conversions in [5XLOOPS[105X (starting with
  version  2.1.0),  using functions of the type [10XIntoSomething([3XX[103X[10X)[110X. The two main
  features that distinguish [10XIntoSomething[110X from [10XAsSomething[110X are:[133X
  
  [30X    [33X[0;6YThe  function  [10XIntoSomething([3XX[103X[10X)[110X  does  not necessarily return the same
        domain  as  [3XX[103X.  The  reason  is  that  [3XX[103X can be a group, for instance,
        defined on one of many possible domains, while [10XIntoLoop([3XX[103X[10X)[110X must result
        in  a loop, and hence be defined on a subset of some interval [22X1[122X, [22Xdots[122X,
        [22Xn[122X (see Section [14X6.1[114X).[133X
  
  [30X    [33X[0;6YIn  some  special  situations, the function [10XIntoSomething([3XX[103X[10X)[110X allows to
        convert  [3XX[103X  into  [10XSomething[110X  even  though  [3XX[103X  does  not  have  all the
        properties of [10XSomething[110X. For instance, every quasigroup is isotopic to
        a  loop, so it makes sense to allow the conversion [10XIntoLoop([3XQ[103X[10X)[110X even if
        the quasigroup [3XQ[103X does not posses a neutral element.[133X
  
  [33X[0;0YDetails of all conversions in [5XLOOPS[105X can be found in Section [14X4.10[114X.[133X
  
  
  [1X3.3 [33X[0;0YCalculating with Quasigroups[133X[101X
  
  [33X[0;0YAlthough  the  quasigroups  are  ultimately  represented  by  multiplication
  tables,  the  algorithms  are  efficient because nearly all calculations are
  delegated  to  groups.  The  connection  between  quasigroups  and groups is
  facilitated  via translations (see Section [14X2.2[114X), and we illustrate it with a
  few examples:[133X
  
  [33X[0;0Y[12XExample:[112X  This example shows how properties of quasigroups can be translated
  into  properties  of  translations  in  a  straightforward  way.  Let [22XQ[122X be a
  quasigroup.  We  ask  if [22XQ[122X is associative. We can either test if [22X(xy)z=x(yz)[122X
  for  every  [22Xx[122X,  [22Xy[122X, [22Xz[122X in [22XQ[122X, or we can ask if [22XL_xy=L_xL_y[122X for every [22Xx[122X, [22Xy[122X in [22XQ[122X.
  Note that since [22XL_xy[122X, [22XL_x[122X and [22XL_y[122X are elements of a permutation group, we do
  not  have  to  refer  directly  to  the  multiplication  table once the left
  translations of [22XQ[122X are known.[133X
  
  [33X[0;0Y[12XExample:[112X  This  example shows how properties of loops can be translated into
  properties  of  translations  in a way that requires some theory. A [13Xleft Bol
  loop[113X is a loop satisfying [22Xx(y(xz)) = (x(yx))z[122X. We claim (without proof) that
  a  loop  [22XQ[122X  is  left  Bol if and only if [22XL_xL_yL_x[122X is a left translation for
  every [22Xx[122X, [22Xy[122X in [22XQ[122X.[133X
  
  [33X[0;0Y[12XExample:[112X  This  example  shows  that  many properties of loops become purely
  group-theoretical  once  they are expressed in terms of translations. A loop
  is [13Xsimple[113X if it has no nontrivial congruences. It is possible to show that a
  loop  is  simple  if  and  only  if  its multiplication group is a primitive
  permutation group.[133X
  
  [33X[0;0YThe main idea of the package is therefore to:[133X
  
  [30X    [33X[0;6Ycalculate  the translations and the associated permutation groups when
        they are needed,[133X
  
  [30X    [33X[0;6Ystore them as attributes,[133X
  
  [30X    [33X[0;6Yuse them in algorithms as often as possible.[133X
  
  
  [1X3.4 [33X[0;0YNaming, Viewing and Printing Quasigroups and their Elements[133X[101X
  
  [33X[0;0Y[5XGAP[105X displays information about objects in two modes: the [10XView[110X mode (default,
  short), and the [10XPrint[110X mode (longer). Moreover, when the name of an object is
  set, the name is always shown, no matter which display mode is used.[133X
  
  [33X[0;0YOnly  loops contained in the libraries of [5XLOOPS[105X are named. For instance, the
  loop  obtained  via  [10XMoufangLoop(32,4)[110X, the 4th Moufang loop of order 32, is
  named "Moufang loop 32/4'' and is shown as [10X<Moufang loop 32/4>[110X.[133X
  
  [33X[0;0YA  generic  quasigroup  of  order [22Xn[122X is displayed as [10X<quasigroup of order n>[110X.
  Similarly, a loop of order [22Xn[122X appears as [10X<loop of order n>[110X.[133X
  
  [33X[0;0YThe  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  [10X<left  alternative  loop  of  order  12>[110X  until  a  stronger property is
  obtained.  Which  property is diplayed is governed by the filters built into
  [5XLOOPS[105X (see Appendix [14XB[114X).[133X
  
  
  [1X3.4-1 [33X[0;0YSetQuasigroupElmName and SetLoopElmName[133X[101X
  
  [29X[2XSetQuasigroupElmName[102X( [3XQ[103X, [3Xname[103X ) [32X function
  [29X[2XSetLoopElmName[102X( [3XQ[103X, [3Xname[103X ) [32X function
  
  [33X[0;0YThe  above  functions  change  the  names of elements of a quasigroup (resp.
  loop) [3XQ[103X to [3Xname[103X.[133X
  
  [33X[0;0YBy  default,  elements  of  a quasigroup appear as [10Xqi[110X and elements of a loop
  appear  as  [10Xli[110X  in  both  display  modes, where [10Xi[110X is a positive integer. The
  neutral element of a loop is always indexed by 1.[133X
  
  [33X[0;0YFor  quasigroups  and loops in the [10XPrint[110X mode, we display the multiplication
  table (if it is known), otherwise we display the elements.[133X
  
  [33X[0;0YIn the following example, [10XL[110X is a loop with two elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL;
[127X[104X
    [4X[28X<loop of order 2>
[128X[104X
    [4X[25Xgap>[125X [27XPrint( L );
[127X[104X
    [4X[28X<loop with multiplication table [ [ 1,  2 ], [  2,  1 ] ]>
[128X[104X
    [4X[25Xgap>[125X [27XElements( L );
[127X[104X
    [4X[28X[ l1, l2 ]
[128X[104X
    [4X[25Xgap>[125X [27XSetLoopElmName( L, "loop_element" );; Elements( L );
[127X[104X
    [4X[28X[ loop_element1, loop_element2 ]
[128X[104X
  [4X[32X[104X
  
