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############################################################################
##
#W grpgraph.gd GAP4 package `groupoids' Chris Wensley
#W & Emma Moore
##
## This file contains the declarations for involutory FpWeightedDigraphs
## and FpWeightedDigraphs of groups,
## and normal forms for FpWeightedDigraph of groups.
##############################################################################
##
#C IsGroupoidDigraph( <dig> )
##
DeclareCategory( "IsGroupoidDigraph", IsDomain );
#############################################################################
##
#O GroupoidVertices( <dig> );
#O GroupoidArcs( <dig> );
##
## Vertices must be declared as Operation, and not Attribute,
## so as not to conflict with Vertices in the Grape package.
##
DeclareOperation( "GroupoidVertices", [ IsGroupoidDigraph ] );
DeclareOperation( "GroupoidArcs", [ IsGroupoidDigraph ] );
##############################################################################
##
#R IsFpWeightedDigraphRep( <dig> )
#V IsFpWeightedDigraphFamily( <dig> )
#T IsFpWeightedDigraphType( <dig> )
#P IsFpWeightedDigraph( <dig> )
##
## A FpWeightedDigraph is a set of vertices and a set of directed arcs
##
DeclareRepresentation( "IsFpWeightedDigraphRep",
IsGroupoidDigraph and IsComponentObjectRep,
[ "group", "vertices", "arcs" ] );
DeclareProperty( "IsFpWeightedDigraph", IsGroupoidDigraph );
BindGlobal( "IsFpWeightedDigraphFamily",
NewFamily( "IsFpWeightedDigraphFamily", IsFpWeightedDigraphRep ) );
BindGlobal( "IsFpWeightedDigraphType",
NewType( IsFpWeightedDigraphFamily, IsFpWeightedDigraphRep ) );
#############################################################################
##
#O FpWeightedDigraphNC( <group>, <vertices>, <arcs> )
#O FpWeightedDigraph( <group>, <vertices>, <arcs> )
##
DeclareOperation( "FpWeightedDigraphNC",
[ IsGroup, IsHomogeneousList, IsHomogeneousList ] );
DeclareOperation( "FpWeightedDigraph",
[ IsGroup, IsHomogeneousList, IsHomogeneousList ] );
##############################################################################
##
#A InvolutoryArcs( <dig> )
##
## An involutory digraph is a set of vertices and a set of directed arcs
## and an involution on the set of arcs which reverses source and target
##
DeclareAttribute( "InvolutoryArcs", IsFpWeightedDigraph );
#############################################################################
##
#O FpWeightedAdjacencyMatrix( <dig> )
#O ArcsIsosFromMatrices( <vertices>, <wt_adj_mx>, <isos_mx> )
DeclareOperation( "FpWeightedAdjacencyMatrix", [ IsGroupoidDigraph ] );
DeclareOperation( "ArcsIsosFromMatrices",
[ IsHomogeneousList, IsHomogeneousList, IsHomogeneousList ] );
## ------------------------------------------------------------------------##
## Graphs of Groups ##
## ------------------------------------------------------------------------##
##############################################################################
##
#P IsStructuredDigraph( <dig> )
##
DeclareProperty( "IsStructuredDigraph", IsGroupoidDigraph );
#############################################################################
##
#C IsGraphOfGroups( <gg> )
#R IsGraphOfGroupsRep( <gg> )
#V IsGraphOfGroupsFamily
#T IsGraphOfGroupsType
##
## A FpWeightedDigraph of groups is a 4-tuple containing
## - a FpWeightedDigraph,
## - groups associated with each vertex,
## - subgroup associated to tail vertex of each edge, and
## - an isomorphism associated to each edge.
##
DeclareCategory( "IsGraphOfGroups", IsGroupoidDigraph );
DeclareRepresentation( "IsGraphOfGroupsRep",
IsStructuredDigraph and IsAttributeStoringRep,
[ "DigraphOfGraphOfGroups", "GroupsOfGraphOfGroups",
#? "SubgroupsOfGraphOfGroups",
"IsomorphismsOfGraphOfGroups" ] );
BindGlobal( "IsGraphOfGroupsFamily",
NewFamily( "IsGraphOfGroupsFamily", IsGraphOfGroups ) );
BindGlobal( "IsGraphOfGroupsType",
NewType( IsGraphOfGroupsFamily, IsGraphOfGroupsRep ) );
##############################################################################
##
#P IsGraphOfPermGroups( <gg> )
#P IsGraphOfFpGroups( <gg> )
#P IsGraphOfPcGroups( <gg> )
##
DeclareProperty( "IsGraphOfPermGroups", IsGraphOfGroups );
DeclareProperty( "IsGraphOfFpGroups", IsGraphOfGroups );
DeclareProperty( "IsGraphOfPcGroups", IsGraphOfGroups );
#############################################################################
##
#O GraphOfGroupsNC( <dig>, <gps>, <isos> )
#O GraphOfGroups( <dig>, <gps>, <isos> )
##
DeclareOperation( "GraphOfGroupsNC",
[ IsFpWeightedDigraph, IsList, IsList ] );
DeclareOperation( "GraphOfGroups",
[ IsFpWeightedDigraph, IsList, IsList ] );
#############################################################################
##
#A GroupsOfGraphOfGroups( <gg> )
#A DigraphOfGraphOfGroups( <gg> )
#A SubgroupsOfGraphOfGroups( <gg> )
#A IsomorphismsOfGraphOfGroups( <gg> )
#A RightTransversalsOfGraphOfGroups( <gg> )
#A LeftTransversalsOfGraphOfGroups( <gg> )
##
DeclareAttribute( "GroupsOfGraphOfGroups", IsGraphOfGroups );
DeclareAttribute( "DigraphOfGraphOfGroups", IsGraphOfGroups );
## DeclareAttribute( "SubgroupsOfGraphOfGroups", IsGraphOfGroups );
DeclareAttribute( "IsomorphismsOfGraphOfGroups", IsGraphOfGroups );
DeclareAttribute( "RightTransversalsOfGraphOfGroups", IsGraphOfGroups );
DeclareAttribute( "LeftTransversalsOfGraphOfGroups", IsGraphOfGroups );
## ------------------------------------------------------------------------##
## Rewriting Functions ##
## ------------------------------------------------------------------------##
#############################################################################
##
#A FreeSemigroupOfKnuthBendixRewritingSystem( <kbrws> )
#O NormalFormKBRWS( <group>, <word> )
##
## creates inverse to iso = IsomorphismFpSemigroup
##
DeclareAttribute( "FreeSemigroupOfKnuthBendixRewritingSystem",
IsKnuthBendixRewritingSystem );
DeclareOperation( "NormalFormKBRWS", [ IsFpGroup, IsObject ] );
#############################################################################
##
#O FreeProductWithAmalgamation( <grp>, <grp>, <iso> )
#O FreeProductWithAmalgamationOp( <grp>, <grp>, <iso> )
#A FreeProductWithAmalgamationInfo( <fpagrp> )
#P IsFreeProductWithAmalgamation( <fpgrp> )
##
DeclareOperation( "FreeProductWithAmalgamation",
[ IsGroup, IsGroup, IsGroupHomomorphism ] );
DeclareOperation( "FreeProductWithAmalgamationOp",
[ IsGroup, IsGroup, IsGroupHomomorphism ] );
DeclareProperty( "IsFreeProductWithAmalgamation", IsFpGroup );
DeclareAttribute( "FreeProductWithAmalgamationInfo",
IsFreeProductWithAmalgamation, "mutable" );
#############################################################################
##
#O HnnExtension( <grp>, <iso> )
#P IsHnnExtension( <fpgrp> )
#A HnnExtensionInfo( <fpa> )
##
DeclareOperation( "HnnExtension", [ IsGroup, IsGroupHomomorphism ] );
DeclareProperty( "IsHnnExtension", IsFpGroup );
DeclareAttribute( "HnnExtensionInfo", IsHnnExtension, "mutable" );
#############################################################################
##
#A GraphOfGroupsRewritingSystem( <fpgrp> )
#O NormalFormGGRWS( <fpgrp>, <word> )
##
DeclareAttribute( "GraphOfGroupsRewritingSystem", IsFpGroup );
DeclareOperation( "NormalFormGGRWS", [ IsFpGroup, IsObject ] );
## ------------------------------------------------------------------------##
## Graph of Groups Words ##
## ------------------------------------------------------------------------##
##############################################################################
##
#R IsGraphOfGroupsWordRep( <ggword> )
#V IsGraphOfGroupsWordFamily( <ggword> )
#T IsGraphOfGroupsWordType( <ggword> )
##
## A GraphOfGroupsWord is a word made from elements in the group
## and arcs in the digraph
##
DeclareRepresentation( "IsGraphOfGroupsWordRep",
## IsGroupoidElement and IsAttributeStoringRep,
IsObject and IsAttributeStoringRep,
[ "GraphOfGroupsOfWord",
"TailOfGraphOfGroupsWord", "WordOfGraphOfGroupsWord" ] );
BindGlobal( "IsGraphOfGroupsWordFamily",
NewFamily( "IsGraphOfGroupsWordFamily",
IsGraphOfGroupsWordRep ) );
BindGlobal( "IsGraphOfGroupsWordType",
NewType( IsGraphOfGroupsWordFamily, IsGraphOfGroupsWordRep ) );
##############################################################################
##
#P IsGraphOfGroupsWord( <ggword> )
#P IsReducedGraphOfGroupsWord( <ggword> )
## A GraphOfGroupsWord is Reduced if of the form [ t, h, t, h, ..., t, g ]
##
DeclareProperty( "IsGraphOfGroupsWord", IsGraphOfGroupsWordRep );
DeclareProperty( "IsReducedGraphOfGroupsWord", IsGraphOfGroupsWord );
#############################################################################
##
#O GraphOfGroupsWordNC( <gg>, <tv>, <wL> )
#O GraphOfGroupsWord( <gg>, <tv>, <wL> )
##
DeclareOperation( "GraphOfGroupsWordNC", [ IsGraphOfGroups, IsInt, IsList ] );
DeclareOperation( "GraphOfGroupsWord", [ IsGraphOfGroups, IsInt, IsList ] );
#############################################################################
##
#A GraphOfGroupsOfWord( <ggword> )
#A HeadOfGraphOfGroupsWord( <ggword> )
#A TailOfGraphOfGroupsWord( <ggword> )
#A WordOfGraphOfGroupsWord( <ggword> )
#O ReducedGraphOfGroupsWord( <ggword> )
##
DeclareAttribute( "GraphOfGroupsOfWord", IsGraphOfGroupsWordRep );
DeclareAttribute( "HeadOfGraphOfGroupsWord", IsGraphOfGroupsWordRep );
DeclareAttribute( "TailOfGraphOfGroupsWord", IsGraphOfGroupsWordRep );
DeclareAttribute( "WordOfGraphOfGroupsWord", IsGraphOfGroupsWordRep);
DeclareOperation( "ReducedGraphOfGroupsWord", [ IsGraphOfGroupsWordRep ] );
#############################################################################
##
#P IsMappingToGroupWithGGRWS( <map> )
#O ReducedImageElm( <hom>, <elm> )
##
DeclareProperty( "IsMappingToGroupWithGGRWS",
IsGroupGeneralMappingByImages );
DeclareOperation( "ReducedImageElm", [ IsMappingToGroupWithGGRWS,
IsMultiplicativeElementWithInverse ] );
## ------------------------------------------------------------------------##
## Graph of Groups Groupoid - still be to implemented ##
## ------------------------------------------------------------------------##
#############################################################################
##
## #P IsGraphOfGroupsGroupoid( <gpd> )
## #O GraphOfGroupsGroupoid( <gg> )
##
## DeclareProperty( "IsGraphOfGroupsGroupoid", IsGroupoid );
## DeclareOperation( "GraphOfGroupsGroupoid", [ IsGraphOfGroups ] );
[ Dauer der Verarbeitung: 0.17 Sekunden
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