174 lines
7.9 KiB
GDScript3
174 lines
7.9 KiB
GDScript3
#############################################################################
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##
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#W quasigroups.gd Representing, creating and displaying quasigroups [loops]
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##
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#H @(#)$Id: quasigroups.gd, v 3.4.0 2017/10/17 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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## GAP CATEGORIES AND REPRESENTATIONS
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## -------------------------------------------------------------------------
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## Categories convenient for defining quasigroups
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## element which is an admissible argument for the right argument of /
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DeclareCategory( "IsRightQuotientElement", IsExtLElement);
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DeclareCategoryCollections("IsRightQuotientElement");
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DeclareCategoryCollections("IsRightQuotientElementCollection");
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## Every associative element with an inverse can form right quotients
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## (in fact, in some sense it might be enough to have just a left inverse,
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## but there doesn't seem to be any benefit to delving to that level of
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## detail at this point.)
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## By noting this property, we can create a RightQuasigroup from, e.g., group
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## elements
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InstallTrueMethod(IsRightQuotientElement,
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IsMultiplicativeElementWithInverse and IsAssociativeElement);
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## Now what we would like to do is re-declare
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## DeclareOperation( "/", [IsExtRElement, IsRightQuotientElement] );
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## but we can't since "/" is in the kernel, so we will have to content
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## ourselves with InstallOtherMethod() calls on /. (I am not actually sure what
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## the practical upshot of that is, i.e. if it has any shortcomings as compared
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## to if we could declare "/" more generally.)
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## Element which is admissible for the left argument of LeftQuotient()
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DeclareCategory( "IsLeftQuotientElement", IsExtRElement);
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DeclareCategoryCollections("IsLeftQuotientElement");
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DeclareCategoryCollections("IsLeftQuotientElementCollection");
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## Every associative element with an inverse can form left quotients
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InstallTrueMethod(IsLeftQuotientElement,
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IsMultiplicativeElementWithInverse and IsAssociativeElement);
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## Again, ideally (in some sense) we'd like to redeclare
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## DeclareOperation("LeftQuotient", [IsLeftQuotientElement,IsExtLElement]);
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## element of a quasigroup
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DeclareSynonym( "IsQuasigroupElement",
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IsMultiplicativeElement and
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IsLeftQuotientElement and IsRightQuotientElement );
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DeclareRepresentation( "IsQuasigroupElmRep",
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IsPositionalObjectRep and IsQuasigroupElement, [1] );
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## element of a loop
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DeclareSynonym( "IsLoopElement",
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IsQuasigroupElement and IsMultiplicativeElementWithInverse );
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DeclareRepresentation( "IsLoopElmRep",
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IsQuasigroupElmRep and IsMultiplicativeElementWithInverse, [1] );
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## Right quasigroup
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DeclareCategory("IsRightQuasigroup",
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IsMagma and IsRightQuotientElementCollection);
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## Although the following assertion is mathematically correct, unfortunately
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## it interferes with method selection for standard group operations
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## in GAP. As an example, if it is uncommented, it will no longer be possible
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## to construct a CyclicGroup; trying to do so eventually dies in
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## GeneratorsOfRightQuasigroup. Those errors could conceivably be corrected by
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## delving further into GAP's method selection mechanism and adjusting the
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## declarations of various quasigroup operations, but it doesn't seem worth
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## the effort as there is unlikely to be much call to consider a group as a
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## quasigroup. If it is desirable to do so in a particular case, it should be
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## possible to use the elements of the group to form a quasigroup, since they
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## will all satisfy IsRightQuotientElement by a TrueMethod installed above.
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## InstallTrueMethod(IsRightQuasigroup, IsGroup);
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## Left quasigroup
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DeclareCategory("IsLeftQuasigroup",
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IsMagma and IsLeftQuotientElementCollection);
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## We forego the following for the reasons outlined above for right quasigroups.
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## InstallTrueMethod(IsLeftQuasigroup, IsGroup);
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## quasigroup
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DeclareSynonym( "IsQuasigroup", IsRightQuasigroup and IsLeftQuasigroup );
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## loop
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DeclareSynonym( "IsLoop", IsQuasigroup and IsMagmaWithOne and
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IsMultiplicativeElementWithInverseCollection);
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#############################################################################
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## TESTING MULTIPLICATION TABLES
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## -------------------------------------------------------------------------
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DeclareProperty( "IsLeftQuasigroupTable", IsMatrix );
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DeclareProperty( "IsRightQuasigroupTable", IsMatrix );
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DeclareSynonym( "IsQuasigroupTable",
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IsLeftQuasigroupTable and IsRightQuasigroupTable );
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DeclareSynonym( "IsQuasigroupCayleyTable", IsQuasigroupTable );
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DeclareProperty( "IsLoopTable", IsMatrix );
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DeclareSynonym( "IsLoopCayleyTable", IsLoopTable );
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DeclareGlobalFunction("CanonicalCayleyTableOfLeftQuasigroupTable");
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DeclareOperation( "CanonicalCayleyTable", [ IsMatrix ] );
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DeclareOperation( "NormalizedQuasigroupTable", [ IsMatrix ] );
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS MANUALLY
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## -------------------------------------------------------------------------
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DeclareAttribute( "CayleyTable", IsMagma );
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DeclareOperation( "QuasigroupByCayleyTable", [ IsMatrix ] );
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DeclareOperation( "LoopByCayleyTable", [ IsMatrix ] );
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DeclareOperation( "SpecifyElmNamePrefix", [ IsCollection, IsString ] );
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DeclareSynonym( "SetQuasigroupElmName", SpecifyElmNamePrefix );
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DeclareSynonym( "SetLoopElmName", SpecifyElmNamePrefix );
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DeclareOperation( "BindElmNames", [ IsMagma ] );
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DeclareAttribute( "ConstructorFromTable", IsMagma );
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DeclareOperation( "CanonicalCopy", [ IsMagma ] );
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS FROM A FILE
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## -------------------------------------------------------------------------
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DeclareOperation( "QuasigroupFromFile", [ IsString, IsString ] );
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DeclareOperation( "LoopFromFile", [ IsString, IsString ] );
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#############################################################################
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## CREATING QUASIGROUPS AND LOOPS BY SECTIONS
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## -------------------------------------------------------------------------
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DeclareGlobalFunction("CayleyTableByPerms");
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DeclareOperation( "QuasigroupByLeftSection", [ IsPermCollection ] );
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DeclareOperation( "LoopByLeftSection", [ IsPermCollection ] );
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DeclareOperation( "QuasigroupByRightSection", [ IsPermCollection ] );
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DeclareOperation( "LoopByRightSection", [ IsPermCollection ] );
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DeclareOperation( "QuasigroupByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
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DeclareOperation( "LoopByRightFolder", [ IsGroup, IsGroup, IsMultiplicativeElementCollection ] );
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#############################################################################
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## CONVERSIONS
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## -------------------------------------------------------------------------
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DeclareOperation( "IntoQuasigroup", [ IsMagma ] );
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DeclareOperation( "PrincipalLoopIsotope",
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[ IsQuasigroup, IsQuasigroupElement, IsQuasigroupElement ] );
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DeclareOperation( "IntoLoop", [ IsMagma ] );
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DeclareOperation( "IntoGroup", [ IsMagma ] );
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#############################################################################
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## PRODUCTS OF QUASIGROUPS AND LOOPS
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## --------------------------------------------------------------------------
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DeclareGlobalFunction("ProductTableOfCanonicalCayleyTables");
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#DirectProduct already declared for groups.
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#############################################################################
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## OPPOSITE QUASIGROUPS AND LOOPS
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## --------------------------------------------------------------------------
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DeclareGlobalFunction( "OppositeQuasigroup");
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DeclareGlobalFunction( "OppositeLoop");
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DeclareAttribute( "Opposite", IsMagma );
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#############################################################################
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## AUXILIARY
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## --------------------------------------------------------------------------
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DeclareGlobalFunction( "LOOPS_ReadCayleyTableFromFile" );
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DeclareGlobalFunction( "LOOPS_CayleyTableByRightFolder" );
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