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##
#W gpd.gd GAP4 package `groupoids' Chris Wensley
#W & Emma Moore
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##
#O RootGroup( <gpd> )
#A RaysOfGroupoid( <gpd> )
#O RayArrowsOfGroupoid( <gpd> )
#A GeneratorsOfGroupoid( <gpd> )
##
DeclareOperation( "RootGroup", [ IsGroupoid and IsSinglePiece ] );
DeclareAttribute( "RaysOfGroupoid", IsGroupoid );
DeclareOperation( "RayArrowsOfGroupoid", [ IsGroupoid ] );
DeclareAttribute( "GeneratorsOfGroupoid", IsGroupoid );
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##
#R IsSinglePieceRaysRep( <gpd> )
#T IsSinglePieceRaysType( <gpd> )
#A LargerDirectProductGroupoid( <gpd> )
##
## A connected groupoid with variable object groups is a wide subgroupoid
## of an IsDirectProductWithCompleteDigraph with the same objects, and is
## determined by a root group, and a set of conjugating elements
##
DeclareRepresentation( "IsSinglePieceRaysRep",
IsGroupoid and IsAttributeStoringRep and IsComponentObjectRep,
[ "parent", "rootGroup", "rays" ] );
BindGlobal( "IsSinglePieceRaysType",
NewType( IsGroupoidFamily, IsSinglePieceRaysRep ) );
DeclareAttribute( "LargerDirectProductGroupoid", IsSinglePieceRaysRep );
#############################################################################
##
#P IsPermGroupoid( <gpd> )
#P IsFpGroupoid( <gpd> )
#P IsPcGroupoid( <gpd> )
#P IsMatrixGroupoid( <gpd> )
#P IsFreeGroupoid( <gpd> )
#P IsSinglePieceGroupoidWithRays( <gpd> )
##
DeclareProperty( "IsPermGroupoid", IsGroupoid );
DeclareProperty( "IsFpGroupoid", IsGroupoid );
DeclareProperty( "IsPcGroupoid", IsGroupoid );
DeclareProperty( "IsMatrixGroupoid", IsGroupoid );
DeclareProperty( "IsFreeGroupoid", IsGroupoid );
DeclareProperty( "IsSinglePieceGroupoidWithRays", IsGroupoid );
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##
#F Groupoid( <args> )
##
DeclareGlobalFunction( "Groupoid" );
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##
#O SinglePieceGroupoidNC( <group>, <objects> )
#O SinglePieceGroupoid( <group>, <objects> )
#O GroupoidWithRays( <rootgp>, <objects>, <rays> )
#O SubgroupoidWithRaysNC( <parent>, <rootgp>, <rays> )
#O SubgroupoidWithRays( <parent>, <rootgp>, <rays> )
#O SinglePieceSubgroupoidByGenerators( <ancestor>, <gens> )
#O SinglePieceGroupoidWithRaysNC( <group>, <objects>, <rays> )
#O SinglePieceGroupoidWithRays( <group>, <objects>, <rays> )
##
DeclareOperation( "SinglePieceGroupoidNC",
[ IsGroup, IsHomogeneousList ] );
DeclareOperation( "SinglePieceGroupoid",
[ IsGroup, IsHomogeneousList ] );
DeclareOperation( "GroupoidWithRays",
[ IsGroup, IsHomogeneousList, IsList ] );
DeclareOperation( "SubgroupoidWithRaysNC",
[ IsGroupoid, IsGroup, IsHomogeneousList ] );
DeclareOperation( "SubgroupoidWithRays",
[ IsGroupoid, IsGroup, IsHomogeneousList ] );
DeclareOperation( "SinglePieceSubgroupoidByGenerators",
[ IsGroupoid, IsList ] );
DeclareOperation( "SinglePieceGroupoidWithRaysNC",
[ IsGroup, IsHomogeneousList, IsHomogeneousList ] );
DeclareOperation( "SinglePieceGroupoidWithRays",
[ IsGroup, IsHomogeneousList, IsHomogeneousList ] );
#############################################################################
##
#O GroupoidByIsomorphisms( <rootgp>, <objects>, <isos> )
#P IsGroupoidByIsomorphisms( <gpd> )
##
DeclareOperation( "GroupoidByIsomorphisms",
[ IsGroup, IsHomogeneousList, IsList ] );
DeclareProperty( "IsGroupoidByIsomorphisms",
IsSinglePieceDomain and IsGroupoid );
#############################################################################
##
#O ObjectGroup( <gpd>, <obj> )
#A ObjectGroups( <gpd> )
##
DeclareOperation( "ObjectGroup", [ IsGroupoid, IsObject ] );
DeclareAttribute( "ObjectGroups", IsGroupoid );
## ======================================================================== ##
## Homogeneous groupoids ##
## ======================================================================== ##
##############################################################################
##
#P IsHomogeneousDiscreteGroupoid( <gpd> )
#R IsHomogeneousDiscreteGroupoidRep( <gpd> )
#T IsHomogeneousDiscreteGroupoidType( <gpd> )
##
## A homogeneous, discrete groupoid is a union of identical, single domain
## groupoids - just the thing for the source of an xmod of groupoids!
## So such structures deserve their own representation and type.
##
DeclareSynonymAttr( "IsHomogeneousDiscreteGroupoid", IsGroupoid and
IsDiscreteDomainWithObjects and IsHomogeneousDomainWithObjects );
DeclareRepresentation( "IsHomogeneousDiscreteGroupoidRep",
IsGroupoid and IsAttributeStoringRep and IsComponentObjectRep,
[ "magma", "objects" ] );
BindGlobal( "IsHomogeneousDiscreteGroupoidType",
NewType( IsGroupoidFamily,
IsHomogeneousDiscreteGroupoidRep and IsGroupoid ) );
##############################################################################
##
#O HomogeneousGroupoid( <gpd>, <obs> )
#O HomogeneousGroupoidNC( <gpd>, <obs> )
#O HomogeneousDiscreteGroupoid( <gp>, <obs> )
##
DeclareOperation( "HomogeneousGroupoid", [ IsGroupoid, IsHomogeneousList ] );
DeclareOperation( "HomogeneousGroupoidNC", [ IsGroupoid, IsHomogeneousList ] );
DeclareOperation( "HomogeneousDiscreteGroupoid",
[ IsGroup, IsHomogeneousList ] );
## ======================================================================== ##
## Manipulating groupoid unions ##
## ======================================================================== ##
##############################################################################
##
#O ReplaceOnePieceInUnion( <union>, <old>, <new> )
##
DeclareOperation( "ReplaceOnePieceInUnion",
[ IsGroupoid and IsPiecesRep, IsGroupoid and IsSinglePiece,
IsGroupoid and IsSinglePiece ] );
DeclareOperation( "ReplaceOnePieceInUnion",
[ IsGroupoid and IsPiecesRep, IsPosInt,
IsGroupoid and IsSinglePiece ] );
## ======================================================================== ##
## Groupoid Elements ##
## ======================================================================== ##
##############################################################################
##
#O GroupoidElement( <gpd>, <elt>, <src>, <tgt> )
#O IdentityArrow( <gpd>, <obj> )
##
## A connected groupoid element is [ group element, tail, head ]
##
DeclareSynonym( "GroupoidElement", Arrow );
DeclareOperation( "IdentityArrow", [ IsGroupoid, IsObject ] );
##############################################################################
##
## Stars, Costars and Homsets are particular subsets of the elements
## of a connected groupoid for which an iterator is required
##
## Up to seven items of data are required:
## 1. a list of source objects,
## 2. a list of target objects,
## 3. a list of source rays,
## 4. a list of target rays,
## 5. a representative arrow,
## 6. a list of group elements, e.g. an object group of a coset,
## 7. a single character string 'c' in {s,c,h,u,r,l,d},
## specifying star; costar; homset; union; right/left/double coset.
##
#P IsHomsetCosets( <obj> )
#R IsHomsetCosetsRep( <obj> )
#R IsHomsetCosetsFamily( <obj> )
#R IsHomsetCosetsType( <obj> )
#O ObjectStarNC( <gpd>, <obj> )
#O ObjectStar( <gpd>, <obj> )
#O ObjectCostarNC( <gpd>, <obj> )
#O ObjectCostar( <gpd>, <obj> )
#O HomsetNC( <gpd>, <obj1>, <obj2> )
#O Homset( <gpd>, <obj1>, <obj2> )
#A ElementsOfGroupoid( <gpd> )
##
DeclareProperty( "IsHomsetCosets", IsGroupoidElementCollection );
InstallTrueMethod(IsListOrCollection, IsHomsetCosets);
DeclareRepresentation( "IsHomsetCosetsRep", IsHomsetCosets
and IsAttributeStoringRep and IsComponentObjectRep,
[ "tobj", "hobj", "trays", "hrays", "rep", "elements", "type" ] );
BindGlobal( "IsHomsetCosetsFamily",
NewFamily( "IsHomsetCosetsFamily", IsList ) );
BindGlobal( "IsHomsetCosetsType",
NewType( CollectionsFamily( IsGroupoidElementFamily ),
IsHomsetCosetsRep ) );
DeclareOperation( "ObjectStarNC", [ IsGroupoid, IsObject ] );
DeclareOperation( "ObjectStar", [ IsGroupoid, IsObject ] );
DeclareOperation( "ObjectCostarNC", [ IsGroupoid, IsObject ] );
DeclareOperation( "ObjectCostar", [ IsGroupoid, IsObject ] );
DeclareOperation( "HomsetNC", [ IsGroupoid, IsObject, IsObject ] );
DeclareOperation( "Homset", [ IsGroupoid, IsObject, IsObject ] );
DeclareAttribute( "ElementsOfGroupoid", IsGroupoid );
##############################################################################
##
#P IsGroupoidCoset( <cset> )
#A SuperDomain( <cset> ); #? rename ??
#O RightCosetRepresentatives( <gpd>, <sgpd> ) #? should be Iterator ??
#R IsLeftCosetWithObjectsDefaultRep( <gp>, <obj> )
#O LeftCosetRepresentatives( <gpd>, <sgpd> ) #? should be Iterator ??
#O LeftCosetRepresentativesFromObject( <gpd>, <sgpd>, <obj> )
#O DoubleCosetRepresentatives( <gpd>, <sgpd>, <sgpd> ) #? ditto ??
##
DeclareProperty( "IsGroupoidCoset", IsRightCosetDefaultRep );
DeclareAttribute( "SuperDomain", IsRightCosetDefaultRep );
DeclareCategory( "IsLeftCosetWithObjects", IsDomain and IsExternalOrbit );
DeclareRepresentation( "IsLeftCosetWithObjectsDefaultRep",
IsComponentObjectRep and IsAttributeStoringRep
and IsLeftCosetWithObjects, [] );
DeclareOperation( "RightCosetRepresentatives", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "LeftCosetRepresentatives", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "LeftCosetRepresentativesFromObject",
[ IsGroupoid, IsGroupoid, IsObject ] );
DeclareOperation( "DoubleCosetRepresentatives",
[ IsGroupoid, IsGroupoid, IsGroupoid ] );
## ======================================================================== ##
## Subgroupoids ##
## ======================================================================== ##
#############################################################################
##
#F Subgroupoid( <args> )
#O IsSubgroupoid( <G>, <S> )
#O IsWideSubgroupoid( <G>, <S> )
#O IsFullSubgroupoid( <G>, <S> )
##
DeclareGlobalFunction( "Subgroupoid" );
DeclareOperation( "IsSubgroupoid", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "IsWideSubgroupoid", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "IsFullSubgroupoid", [ IsGroupoid, IsGroupoid ] );
#############################################################################
##
#O SubgroupoidBySubgroup( <gpd>, <sgp> )
#O SubgroupoidByObjects( <gpd>, <obs> )
#O SubgroupoidByPieces( <gpd>, <obhoms> )
#O PiecePositions( <gpd>, <sgpd> )
#O DiscreteSubgroupoid( <gpd>, <gps>, <obs> )
#O HomogeneousDiscreteSubgroupoid( <gpd>, <gp>, <obs> )
#A MaximalDiscreteSubgroupoid( <gpd> )
#A FullTrivialSubgroupoid( <gpd> )
#A DiscreteTrivialSubgroupoid( <gpd> )
##
DeclareOperation( "SubgroupoidBySubgroup",
[ IsGroupoid and IsSinglePiece, IsGroup ] );
DeclareOperation( "SubgroupoidByObjects", [ IsGroupoid, IsHomogeneousList ] );
DeclareSynonym( "FullSubgroupoid", SubgroupoidByObjects );
DeclareOperation( "SubgroupoidByPieces",
[ IsGroupoid, IsList ] );
## DeclareOperation( "PiecePositions", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "DiscreteSubgroupoid",
[ IsGroupoid, IsList, IsHomogeneousList ] );
DeclareOperation( "HomogeneousDiscreteSubgroupoid",
[ IsGroupoid, IsGroup, IsHomogeneousList ] );
DeclareAttribute( "MaximalDiscreteSubgroupoid", IsGroupoid );
DeclareAttribute( "FullTrivialSubgroupoid", IsGroupoid );
DeclareAttribute( "DiscreteTrivialSubgroupoid", IsGroupoid );
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##
#O ConjugateGroupoid( <gpd>, <elt> ) . . . . . . conjugate of <gpd> by <elt>
##
DeclareOperation( "ConjugateGroupoid", [ IsGroupoid, IsGroupoidElement ] );
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##
#O NormalSubgroupoid( <G>, <S> )
#O IsNormalSubgroupoid( <G>, <S> )
##
DeclareOperation( "NormalSubgroupoid", [ IsGroupoid, IsGroupoid ] );
DeclareOperation( "IsNormalSubgroupoid", [ IsGroupoid, IsGroupoid ] );
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##
#P IsGroupoidWithMonoidObjects( <gpd> )
##
DeclareProperty( "IsGroupoidWithMonoidObjects", IsGroupoid );
#############################################################################
##
#A RightActionGroupoid( <M> )
##
DeclareAttribute( "RightActionGroupoid", IsMonoid );
