loops/doc/chap3.txt

  
  3 How the Package Works
  
  The package consists of three complementary components:
  
      the  core  algorithms for quasigroup theoretical notions (see Chapters
        4, 5, 6 and 7),
  
      algorithms  for  specific varieties of loops, mostly for Moufang loops
        (see Chapter 8),
  
      the library of small loops (see Chapter 9).
  
  Although  we  do  not explain the algorithms in detail here, we describe the
  general  philosophy  so  that  users  can  anticipate  the  capabilities and
  behavior of LOOPS.
  
  
  3.1 Representing Quasigroups
  
  Since  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  GAP.  (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 4.1 for more details.)
  
  Consequently, the package is intended primarily for quasigroups and loops of
  small order, say up to 1000.
  
  The  GAP  categories  IsQuasigroupElement,  IsLoopElement,  IsQuasigroup and
  IsLoop are declared in LOOPS as follows:
  
  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);
  
  
  3.2 Conversions between magmas, quasigroups, loops and groups
  
  Whether  an  object  is  considered  a magma, quasigroup, loop or group is a
  matter  of  declaration  in  LOOPS. Loops are automatically quasigroups, and
  both  groups  and  quasigroups  are  automatically  magmas. All standard GAP
  commands for magmas are therefore available for quasigroups and loops.
  
  In  GAP,  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 6.1).
  
      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 4.10.
  
  
  3.3 Calculating with Quasigroups
  
  Although  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 2.2), and we illustrate it with a
  few examples:
  
  Example:  This example shows how properties of quasigroups can be translated
  into  properties  of  translations  in  a  straightforward  way.  Let Q be a
  quasigroup.  We  ask  if Q is associative. We can either test if (xy)z=x(yz)
  for  every  x,  y, z in Q, or we can ask if L_xy=L_xL_y for every x, y in Q.
  Note that since L_xy, L_x and L_y are elements of a permutation group, we do
  not  have  to  refer  directly  to  the  multiplication  table once the left
  translations of Q are known.
  
  Example:  This  example shows how properties of loops can be translated into
  properties  of  translations  in a way that requires some theory. A left Bol
  loop is a loop satisfying x(y(xz)) = (x(yx))z. We claim (without proof) that
  a  loop  Q  is  left  Bol if and only if L_xL_yL_x is a left translation for
  every x, y in Q.
  
  Example:  This  example  shows  that  many properties of loops become purely
  group-theoretical  once  they are expressed in terms of translations. A loop
  is simple 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.
  
  The main idea of the package is therefore to:
  
      calculate  the translations and the associated permutation groups when
        they are needed,
  
      store them as attributes,
  
      use them in algorithms as often as possible.
  
  
  3.4 Naming, Viewing and Printing Quasigroups and their Elements
  
  GAP displays information about objects in two modes: the View mode (default,
  short), and the Print mode (longer). Moreover, when the name of an object is
  set, the name is always shown, no matter which display mode is used.
  
  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 B).
  
  
  3.4-1 SetQuasigroupElmName and SetLoopElmName
  
  SetQuasigroupElmName( Q, name )  function
  SetLoopElmName( Q, name )  function
  
  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.
  
    Example  
    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 ]