loops/gap/quasigroups.gd

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2017-10-16 19:43:09 +00:00
#############################################################################
##
#W quasigroups.gd Representing, creating and displaying quasigroups [loops]
##
#H @(#)$Id: quasigroups.gd, v 3.2.0 2016/05/02 gap Exp $
##
#Y Copyright (C) 2004, G. P. Nagy (University of Szeged, Hungary),
#Y P. Vojtechovsky (University of Denver, USA)
##
#############################################################################
## GAP CATEGORIES AND REPRESENTATIONS
## -------------------------------------------------------------------------
## element of a quasigroup
DeclareCategory( "IsQuasigroupElement", IsMultiplicativeElement );
DeclareRepresentation( "IsQuasigroupElmRep",
IsPositionalObjectRep and IsMultiplicativeElement, [1] );
## element of a loop
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 );
## quasigroup
DeclareCategory( "IsQuasigroup", IsMagma and IsLatin );
## loop
DeclareCategory( "IsLoop", IsQuasigroup and IsMultiplicativeElementWithInverseCollection);
#############################################################################
## TESTING MULTIPLICATION TABLES
## -------------------------------------------------------------------------
DeclareOperation( "IsQuasigroupTable", [ IsMatrix ] );
DeclareSynonym( "IsQuasigroupCayleyTable", IsQuasigroupTable );
DeclareOperation( "IsLoopTable", [ IsMatrix ] );
DeclareSynonym( "IsLoopCayleyTable", IsLoopTable );
DeclareOperation( "CanonicalCayleyTable", [ IsMatrix ] );
DeclareOperation( "NormalizedQuasigroupTable", [ IsMatrix ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS MANUALLY
## -------------------------------------------------------------------------
DeclareAttribute( "CayleyTable", IsQuasigroup );
DeclareOperation( "QuasigroupByCayleyTable", [ IsMatrix ] );
DeclareOperation( "LoopByCayleyTable", [ IsMatrix ] );
DeclareOperation( "SetQuasigroupElmName", [ IsQuasigroup, IsString ] );
DeclareSynonym( "SetLoopElmName", SetQuasigroupElmName );
DeclareOperation( "CanonicalCopy", [ IsQuasigroup ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS FROM A FILE
## -------------------------------------------------------------------------
DeclareOperation( "QuasigroupFromFile", [ IsString, IsString ] );
DeclareOperation( "LoopFromFile", [ IsString, IsString ] );
#############################################################################
## CREATING QUASIGROUPS AND LOOPS BY SECTIONS
## -------------------------------------------------------------------------
DeclareOperation( "CayleyTableByPerms", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByLeftSection", [ IsPermCollection ] );
DeclareOperation( "LoopByLeftSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightSection", [ IsPermCollection ] );
DeclareOperation( "LoopByRightSection", [ IsPermCollection ] );
DeclareOperation( "QuasigroupByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
DeclareOperation( "LoopByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
#############################################################################
## CONVERSIONS
## -------------------------------------------------------------------------
DeclareOperation( "IntoQuasigroup", [ IsMagma ] );
DeclareOperation( "PrincipalLoopIsotope",
[ IsQuasigroup, IsQuasigroupElement, IsQuasigroupElement ] );
DeclareOperation( "IntoLoop", [ IsMagma ] );
DeclareOperation( "IntoGroup", [ IsMagma ] );
#############################################################################
## PRODUCTS OF QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
#DirectProduct already declared for groups.
#############################################################################
## OPPOSITE QUASIGROUPS AND LOOPS
## --------------------------------------------------------------------------
DeclareOperation( "OppositeQuasigroup", [ IsQuasigroup ] );
DeclareOperation( "OppositeLoop", [ IsLoop ] );
DeclareAttribute( "Opposite", IsQuasigroup );
#############################################################################
## AUXILIARY
## --------------------------------------------------------------------------
DeclareGlobalFunction( "LOOPS_ReadCayleyTableFromFile" );
DeclareGlobalFunction( "LOOPS_CayleyTableByRightFolder" );