| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/groupoids/lib/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 11.8.2025 mit Größe 11 kB |
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Quelle grpgraph.gd Sprache: unbekannt |
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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: 0.18 Sekunden (vorverarbeitet) ] |
2026-03-28