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#############################################################################
##
#W mwo.gd GAP4 package `groupoids' Chris Wensley
##
## This file contains the declarations of elements, magma, etc., and their
## families in the case of many objects. So each algebraic structure comes
## with a set of objects, and each element $e$ has a tail object $te$ and a
## head object $he$, and multiplies partially if composable: on the left
## with those elements in its family which have head $te$, and on the right
## with those elements of its family which have tail $he$. Non-composable
## elements return $fail$ when multiplied.
##
########################### DOMAIN WITH OBJECTS ###########################
#############################################################################
##
#C IsDomainWithObjects( <obj> ) . . test if object is a domain with objects
##
DeclareCategory( "IsDomainWithObjects", IsDomain );
############################################################################
##
#F DomainWithObjects( <dom>, <obs> )
##
DeclareGlobalFunction( "DomainWithObjects" );
#############################################################################
##
#P IsSinglePieceDomain( <dwo> ) . . . . . . . . . . . . . a connected domain
#P IsDiscreteDomainWithObjects( <dwo> ) . . . . . with at least two objects
#P IsDirectProductWithCompleteDigraphDomain( <dwo> )
#P IsHomogeneousDomainWithObjects( <dwo> )
##
DeclareProperty( "IsSinglePieceDomain", IsDomainWithObjects );
DeclareProperty( "IsDiscreteDomainWithObjects", IsDomainWithObjects );
DeclareProperty( "IsDirectProductWithCompleteDigraphDomain",
IsDomainWithObjects );
DeclareProperty( "IsHomogeneousDomainWithObjects", IsDomainWithObjects );
#############################################################################
##
#A ObjectList( <dwo> )
#A Pieces( <dwo> )
#A PieceIsomorphisms( <dwo> )
#A RootObject( <dwo> )
##
DeclareAttribute( "ObjectList", IsDomainWithObjects );
DeclareAttribute( "Pieces", IsDomainWithObjects );
DeclareAttribute( "PieceIsomorphisms", IsHomogeneousDomainWithObjects );
DeclareAttribute( "RootObject", IsSinglePieceDomain );
#############################################################################
##
#R IsPiecesRep( <dwo> )
##
## A domain with objects is a list of single piece domains
##
DeclareRepresentation( "IsPiecesRep",
IsComponentObjectRep and IsAttributeStoringRep and IsDomainWithObjects,
[ "Pieces", "ObjectList" ] );
#############################################################################
##
#O UnionOfPieces( <pieces> )
#O UnionOfPiecesOp( <pieces>, <dom> )
##
DeclareGlobalFunction( "UnionOfPieces" );
DeclareOperation( "UnionOfPiecesOp", [ IsList, IsDomainWithObjects ] );
#############################################################################
##
#O PieceOfObject( <dwo>, <obj> )
#O PieceNrOfObject( <dwo>, <obj> )
##
DeclareOperation( "PieceOfObject", [ IsDomainWithObjects, IsObject ] );
DeclareOperation( "PieceNrOfObject", [ IsDomainWithObjects, IsObject ] );
#############################################################################
##
#A KindOfDomainWithObjects( <dwo> )
##
DeclareAttribute( "KindOfDomainWithObjects", IsList );
######################## MULT ELTS WITH OBJECTS ###########################
#############################################################################
##
#C IsMultiplicativeElementWithObjects( <elt> )
##
DeclareCategory( "IsMultiplicativeElementWithObjects",
IsMultiplicativeElement );
DeclareCategoryCollections( "IsMultiplicativeElementWithObjects" );
#############################################################################
##
#C IsMultiplicativeElementWithObjectsAndOnes( <elt> )
##
## An identity element at object $o$ is an element $1_o$ which is a left
## identity for $e$ when $te=o$ and a right identity for $e$ when $he=o$.
DeclareCategory( "IsMultiplicativeElementWithObjectsAndOnes",
IsMultiplicativeElementWithObjects );
DeclareCategoryCollections( "IsMultiplicativeElementWithObjectsAndOnes" );
#############################################################################
##
#C IsMultiplicativeElementWithObjectsAndInverses( <elt> )
##
## An element $e$ has inverse $f$ provided $e*f=1_{te}$ and $f*e=1_{he}$.
DeclareCategory( "IsMultiplicativeElementWithObjectsAndInverses",
IsMultiplicativeElementWithObjectsAndOnes );
DeclareCategoryCollections(
"IsMultiplicativeElementWithObjectsAndInverses" );
#############################################################################
##
#C IsGroupoidElement( <elt> )
#C IsGroupoidByIsomorphismsElement( <elt> )
##
DeclareCategory( "IsGroupoidElement",
IsMultiplicativeElementWithObjectsAndInverses );
DeclareCategoryCollections( "IsGroupoidElement" );
DeclareCategory( "IsGroupoidByIsomorphismsElement",
IsMultiplicativeElementWithObjectsAndInverses );
DeclareCategoryCollections( "IsGroupoidByIsomorphismsElement" );
#############################################################################
##
#V IsMultiplicativeElementWithObjectsFamily . . family for elements of mwos
#V IsMultiplicativeElementWithObjectsAndOnesFamily . . . . . . and with ones
#V IsMultiplicativeElementWithObjectsAndInversesFamily . . . . and inverses
#V IsGroupoidElementFamily . . . . . . . . family for elements of groupoids
#T IsMultiplicativeElementWithObjectsType default type for elements of mwos
#T IsGroupoidElementType . . . . . . default type for elements of groupoids
#T IsGroupoidByIsomorphismsElementType . special type for groupoid elements
##
BindGlobal( "IsMultiplicativeElementWithObjectsFamily",
NewFamily( "IsMultiplicativeElementWithObjectsFamily",
IsMultiplicativeElementWithObjects,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "IsMultiplicativeElementWithObjectsAndOnesFamily",
NewFamily( "IsMultiplicativeElementWithObjectsAndOnesFamily",
IsMultiplicativeElementWithObjectsAndOnes,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "IsMultiplicativeElementWithObjectsAndInversesFamily",
NewFamily( "IsMultiplicativeElementWithObjectsAndInversesFamily",
IsMultiplicativeElementWithObjectsAndInverses,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "IsGroupoidElementFamily",
NewFamily( "IsGroupoidElementFamily", IsGroupoidElement,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "IsMultiplicativeElementWithObjectsType",
NewType( IsMultiplicativeElementWithObjectsFamily,
IsMultiplicativeElementWithObjects ) );
BindGlobal( "IsGroupoidElementType",
NewType( IsGroupoidElementFamily, IsGroupoidElement ) );
BindGlobal( "IsGroupoidByIsomorphismsElementType",
NewType( IsGroupoidElementFamily,
IsGroupoidByIsomorphismsElement ) );
########################### MAGMA WITH OBJECTS ############################
#############################################################################
##
#C IsMagmaWithObjects( <dwo> ) . . . . . . . category of magmas with objects
#C IsMagmaWithObjectsAndOnes( <dwo> ) . . . . . . . . . . . . . . . and ones
#C IsMagmaWithObjectsAndInverses( <dwo> ) . . . . . . . . . . . and inverses
#C IsGroupoid( <dwo> ) . . . . . . . . . . . . . . . . . . and all inverses
##
## A *magma with objects* in {\GAP} is a domain $M$ with (not necessarily
## associative) partial mutliplication.
##
DeclareCategory( "IsMagmaWithObjects", IsDomainWithObjects and
IsMultiplicativeElementWithObjectsCollection );
DeclareCategoryCollections( "IsMagmaWithObjects" );
DeclareCategory( "IsSemigroupWithObjects",
IsMagmaWithObjects and IsAssociative );
DeclareCategory( "IsMonoidWithObjects",
IsSemigroupWithObjects and
IsMultiplicativeElementWithObjectsAndOnesCollection );
DeclareCategory( "IsGroupoid", IsMonoidWithObjects and
IsGroupoidElementCollection );
DeclareCategoryCollections( "IsGroupoid" );
#############################################################################
##
#V IsMagmaWithObjectsFamily . . . . . . . . . family for magmas with objects
#V IsSemigroupWithObjectsFamily . . . . . family for semigroups with objects
#V IsMonoidWithObjectsFamily . . . . . . . . family for monoids with objects
#V IsGroupoidFamily . . . . . . . . . . . . . . . . . . family for groupoids
##
IsMagmaWithObjectsFamily := CollectionsFamily(
IsMultiplicativeElementWithObjectsFamily );
IsSemigroupWithObjectsFamily := CollectionsFamily(
IsMultiplicativeElementWithObjectsAndOnesFamily );
IsMonoidWithObjectsFamily := CollectionsFamily(
IsMultiplicativeElementWithObjectsAndInversesFamily );
IsGroupoidFamily := CollectionsFamily( IsGroupoidElementFamily );
#############################################################################
##
#P IsSinglePiece( <mwo> )
#P IsDiscreteMagmaWithObjects( <mwo> )
#P IsDirectProductWithCompleteDigraph( <mwo> )
##
DeclareSynonymAttr( "IsSinglePiece",
IsMagmaWithObjects and IsSinglePieceDomain );
DeclareSynonymAttr( "IsDiscreteMagmaWithObjects",
IsMagmaWithObjects and IsDiscreteDomainWithObjects );
DeclareSynonymAttr( "IsDirectProductWithCompleteDigraph",
IsMagmaWithObjects and IsDirectProductWithCompleteDigraphDomain );
############################################################################
##
#F MagmaWithObjects( <mag>, <obs> )
##
## A standard single piece magma with objects has elements $(e,t,h)$
## where $e$ is an element of <mag> and $t$, $h$ are the tail and head.
## Multiplication is given by $(e,t,h)*(f,h,k) = (e*f,t,k)$.
## Other constructors are possible.
##
DeclareGlobalFunction( "MagmaWithObjects" );
#############################################################################
##
#A GeneratorsOfMagmaWithObjects( <mwo> )
#A GeneratorsOfSemigroupWithObjects( <mwo> )
#A GeneratorsOfMonoidWithObjects( <mwo> )
##
DeclareAttribute( "GeneratorsOfMagmaWithObjects", IsMagmaWithObjects );
DeclareAttribute( "GeneratorsOfSemigroupWithObjects", IsSemigroupWithObjects );
DeclareAttribute( "GeneratorsOfMonoidWithObjects", IsMonoidWithObjects );
#############################################################################
##
## representation and types for groupoids etc with a single piece
##
#R IsMWOSinglePieceRep
#T IsMagmaWithObjectsType ( <mwo> )
#T IsSemigroupWithObjectsType( <swo> )
#T IsMonoidWithObjectsType( <mwo> )
#T IsGroupoidType( <gpd> )
##
DeclareRepresentation( "IsMWOSinglePieceRep",
IsComponentObjectRep and IsAttributeStoringRep and IsMagmaWithObjects,
[ "objects", "magma" ] );
BindGlobal( "IsMagmaWithObjectsType",
NewType( IsMagmaWithObjectsFamily, IsMWOSinglePieceRep ) );
BindGlobal( "IsSemigroupWithObjectsType",
NewType( IsSemigroupWithObjectsFamily,
IsMWOSinglePieceRep and IsSemigroupWithObjects ) );
BindGlobal( "IsMonoidWithObjectsType",
NewType( IsMonoidWithObjectsFamily,
IsMWOSinglePieceRep and IsMonoidWithObjects ) );
BindGlobal( "IsGroupoidType",
NewType( IsGroupoidFamily,
IsMWOSinglePieceRep and IsGroupoid ) );
#############################################################################
##
## types and properties for groupoids etc with several pieces
##
#T IsMWOPiecesType( <dwo> )
#T IsSemigroupWOPiecesType( <dwo> )
#T IsMonoidWOPiecesType( <dwo> )
#T IsGroupoidPiecesType( <dwo> )
##
BindGlobal( "IsMagmaWOPiecesType",
NewType( IsMagmaWithObjectsFamily,
IsPiecesRep and IsMagmaWithObjects ) );
BindGlobal( "IsSemigroupWOPiecesType",
NewType( IsSemigroupWithObjectsFamily,
IsPiecesRep and IsSemigroupWithObjects ) );
BindGlobal( "IsMonoidWOPiecesType",
NewType( IsMonoidWithObjectsFamily,
IsPiecesRep and IsMonoidWithObjects ) );
BindGlobal( "IsGroupoidPiecesType",
NewType( IsGroupoidFamily,
IsPiecesRep and IsGroupoid and IsAssociative ) );
#############################################################################
##
#O SinglePieceMagmaWithObjects( <mag>, <obs> )
#O MagmaWithSingleObject( <mag>, <obj> )
##
DeclareOperation( "SinglePieceMagmaWithObjects", [ IsMagma, IsCollection ] );
DeclareOperation( "MagmaWithSingleObject", [ IsMagma, IsObject ] );
#############################################################################
##
#O Arrow( <mwo>, <elt>, <tail>, <head> )
#O MultiplicativeElementWithObjects( <mwo>, <elt>, <tail>, <head> )
#O ArrowNC( <mwo> <isgpdelt>, <elt>, <tail>, <head> )
##
DeclareOperation( "Arrow",
[ IsMagmaWithObjects, IsMultiplicativeElement, IsObject, IsObject ] );
DeclareSynonym( "MultiplicativeElementWithObjects", Arrow );
DeclareOperation( "ArrowNC",
[ IsMagmaWithObjects, IsBool, IsMultiplicativeElement, IsObject, IsObject ] );
##############################################################################
##
#O ElementOfArrow( <ewo> )
#O TailOfArrow( <ewo> )
#O HeadOfArrow( <ewo> )
#O GroupoidOfArrow( <ewo> )
##
DeclareOperation( "ElementOfArrow", [ IsMultiplicativeElementWithObjects ] );
DeclareOperation( "TailOfArrow", [ IsMultiplicativeElementWithObjects ] );
DeclareOperation( "HeadOfArrow", [ IsMultiplicativeElementWithObjects ] );
DeclareOperation( "GroupoidOfArrow", [ IsMultiplicativeElementWithObjects ] );
################################ SEMIGROUPS ###############################
############################################################################
##
#F SemigroupWithObjects( <mag>, <obs> )
##
## This is a magma with objects where the vertex magmas are semigroups.
##
DeclareGlobalFunction( "SemigroupWithObjects" );
#############################################################################
##
#O SinglePieceSemigroupWithObjects( <sgp>, <obs> )
##
DeclareOperation( "SinglePieceSemigroupWithObjects",
[ IsSemigroup, IsCollection ] );
################################# MONOIDS #################################
############################################################################
##
#F MonoidWithObjects( <mag>, <obs> )
##
## This is a magma with objects where the vertex magmas are monoids.
##
DeclareGlobalFunction( "MonoidWithObjects" );
#############################################################################
##
#O SinglePieceMonoidWithObjects( <mon>, <obs> )
##
DeclareOperation( "SinglePieceMonoidWithObjects",
[ IsMonoid, IsCollection ] );
################################# GROUPS ##################################
## A *group with objects* is a magma with objects where
## the vertex magmas are groups, and every arrow has an inverse,
## and so is a *groupoid* - see file gpd.gd.
################################# SUBDOMAINS ##############################
#############################################################################
##
#O IsSubdomainWithObjects( <D>, <U> )
#F SubdomainWithObjects( <args> )
##
DeclareOperation( "IsSubdomainWithObjects",
[ IsDomainWithObjects, IsDomainWithObjects ] );
DeclareGlobalFunction( "SubdomainWithObjects" );
#############################################################################
##
#O SubdomainByPieces( <dwo>, <comp> )
#O PiecePositions( <dwo>, <sdwo> )
#O DiscreteSubdomain( <dwo>, <obs>, <doms> )
#A MaximalDiscreteSubdomain( <dwo> )
#O FullSubdomain( <dwo>, <obs> )
##
DeclareOperation( "SubdomainByPieces",
[ IsDomainWithObjects, IsHomogeneousList ] );
DeclareOperation( "PiecePositions",
[ IsDomainWithObjects, IsDomainWithObjects ] );
DeclareOperation( "DiscreteSubdomain",
[ IsDomainWithObjects, IsHomogeneousList, IsHomogeneousList ] );
DeclareAttribute( "MaximalDiscreteSubdomain", IsDomainWithObjects );
DeclareOperation( "FullSubdomain", [IsDomainWithObjects,IsHomogeneousList] );
################################ SUBMAGMAS ################################
#############################################################################
##
#R IsSubmagmaWithObjectsTableRep
##
DeclareRepresentation( "IsSubmagmaWithObjectsTableRep",
IsComponentObjectRep and IsAttributeStoringRep and IsMagmaWithObjects,
[ "objects", "magma", "table" ] );
############################################################################
##
#O IsSubmagmaWithObjectsGeneratingTable( <swo>, <A> )
#O SubmagmaWithObjectsElementsTable( <mag>, <A> )
#O SubmagmaWithObjectsByElementsTable( <swo>, <A> )
DeclareOperation( "IsSubmagmaWithObjectsGeneratingTable", [IsMagma,IsList] );
DeclareOperation( "SubmagmaWithObjectsElementsTable", [IsMagma,IsList] );
DeclareOperation( "SubmagmaWithObjectsByElementsTable",
[ IsMagmaWithObjects, IsList ] );
################################ UTILITIES ##################################
#############################################################################
##
#O Ancestor( <dwo> )
##
DeclareOperation( "Ancestor", [ IsDomainWithObjects ] );
############################## DOUBLE GROUPOIDS ###########################
#C IsDoubleGroupoidElement( <elt> )
#V IsDoubleGroupoidElementFamily . . family for elements of double groupoids
#T IdDoubleGroupoidElementType default type for elements of double groupoids
##
DeclareCategory( "IsDoubleGroupoidElement",
IsMultiplicativeElementWithObjectsAndInverses );
DeclareCategoryCollections( "IsDoubleGroupoidElement" );
BindGlobal( "IsDoubleGroupoidElementFamily",
NewFamily( "IsDoubleGroupoidElementFamily", IsDoubleGroupoidElement,
CanEasilySortElements, CanEasilySortElements ) );
BindGlobal( "IsDoubleGroupoidElementType",
NewType( IsDoubleGroupoidElementFamily, IsDoubleGroupoidElement ) );
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