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#W  gp2up.gd                   GAP4 package `XMod'               Chris Wensley
#W                                                                 & Murat Alp
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##  This file contains declarations for UpMappings, Derivations and Sections
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#Y  Copyright (C) 2001-2017, Chris Wensley et al,  
#Y  School of Computer Science, Bangor University, U.K. 

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#C  IsUp2DimensionalMapping( <map> )
#R  IsUp2DimensionalMappingRep( <map> ) 
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##  A section|derivation is determined by a cat1-group|xmod + generator images
##
DeclareCategory( "IsUp2DimensionalMapping", IsGeneralMapping );
DeclareRepresentation( "IsUp2DimensionalMappingRep", 
    IsUp2DimensionalMapping and IsAttributeStoringRep, 
    [ "Object2d", "UpGeneratorImages", "UpHomomorphism", "UpImagePositions" ] );

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#C  IsUp2DimensionalMappingCollection . . . . category of colls of up-2d-maps
#C  IsUp2DimensionalMappingCollColl . . . . . . .  category of colls of colls 
#C  IsUp2DimensionalMappingCollCollColl . . . category of colls, colls, colls
#V  Up2DimensionalMappingFamily . . . . . family for derivations and sections
#T  Up2DimensionalMappingType( <map> ) 
##
DeclareCategoryCollections( "IsUp2DimensionalMapping" );
DeclareCategoryCollections( "IsUp2DimensionalMappingCollection" );
DeclareCategoryCollections( "IsUp2DimensionalMappingCollColl" );
BindGlobal( "Up2DimensionalMappingFamily", 
    NewFamily( "Up2DimensionalMappingFamily", IsUp2DimensionalMapping, 
               CanEasilySortElements, CanEasilySortElements ) ); 
BindGlobal( "Up2DimensionalMappingType", 
            NewType( Up2DimensionalMappingFamily, 
                     IsUp2DimensionalMappingRep ) ); 

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#A  Object2d( <map> )
#A  UpHomomorphism( <map> ) 
#A  UpGeneratorImages( <map> )
#A  UpImagePositions( <map> )
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DeclareAttribute( "Object2d", IsUp2DimensionalMapping ); 
DeclareAttribute( "UpHomomorphism", IsUp2DimensionalMapping ); 
DeclareAttribute( "UpGeneratorImages", IsUp2DimensionalMapping );
DeclareAttribute( "UpImagePositions", IsUp2DimensionalMapping );

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#P  IsDerivation( <map> )
#P  IsSection( <map> )
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DeclareProperty( "IsDerivation", IsUp2DimensionalMapping );
DeclareProperty( "IsSection", IsUp2DimensionalMapping );


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##                               Derivations                                ##
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#O  DerivationByImages                                     sets up the mapping
#O  DerivationByImagesNC                                   sets up the mapping
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DeclareOperation( "DerivationByImages", 
    [ Is2DimensionalDomain, IsHomogeneousList ] );
DeclareOperation( "DerivationByImagesNC", 
    [ Is2DimensionalDomain, IsHomogeneousList ] );
##  usage: DerivationByImages( XM, im, [, true|false ] )

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#O  DerivationImage                  image of  r \in R  by the derivation  chi
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DeclareOperation( "DerivationImage", [ IsDerivation, IsObject ] );

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#O  DerivationBySection  construct an XMod derivation from a cat1-group section
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DeclareOperation( "DerivationBySection", [ IsSection ] );

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#O  PrincipalDerivation              derivation determined by choice of s in S
#A  PrincipalDerivations                         list of principal derivations
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DeclareOperation( "PrincipalDerivation", [ IsXMod, IsObject ] );
DeclareAttribute( "PrincipalDerivations", IsXMod );

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#O  CompositeDerivation                 Whitehead composite of two derivations
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DeclareOperation( "CompositeDerivation", [ IsDerivation, IsDerivation ] );

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#P  IsRegularDerivation                   so an element of the Whitehead group
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DeclareProperty( "IsRegularDerivation", IsDerivation );

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#A  SourceEndomorphism       upmapping determines endomorphism of source group
#A  RangeEndomorphism        upmapping determines endomorphism of range group
#A  Object2dEndomorphism     upmapping determines endomorphism of xmod or cat1
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DeclareAttribute( "SourceEndomorphism", IsUp2DimensionalMapping );
DeclareAttribute( "RangeEndomorphism", IsUp2DimensionalMapping );
DeclareAttribute( "Object2dEndomorphism", IsUp2DimensionalMapping );

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#O  InverseDerivations      Finds all semigroup inverses XJ for derivation Xi
##                                                i.e.  XiXjXi=Xi & XjXiXj=Xj
DeclareOperation( "InverseDerivations", [ IsDerivation ] );

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#O  ListInverseDerivations               List all inverses for each derivation
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DeclareOperation( "ListInverseDerivations", [ IsXMod ] );


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##                                 Sections                                 ##
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#O  SectionByImages                                   sets up GroupHomByImages
#O  SectionByImagesNC                                 sets up GroupHomByImages
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DeclareOperation( "SectionByImages", 
    [ Is2DimensionalDomain, IsGroupHomomorphism ] );
DeclareOperation( "SectionByImagesNC", 
    [ Is2DimensionalDomain, IsGroupHomomorphism ] );
##  usage: SectionByImages( C, im, [, true|false ] )

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#O  SectionByDerivation      the cat1-group section determined by a derivation
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DeclareOperation( "SectionByDerivation", [ IsDerivation ] );

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#O  CompositeSection                       Whitehead composite of two sections
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DeclareOperation( "CompositeSection", [ IsSection, IsSection ] );


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##                      Monoids of Derivations or Sections                 ##
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#P  IsMonoidOfUp2DimensionalMappings( <obj> )
#R  IsMonoidOfUp2DimensionalMappingsObj( <obj> )
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##  An Up2DimensionalMappings record stores images lists and composition table
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DeclareProperty( "IsMonoidOfUp2DimensionalMappings", IsObject );
DeclareRepresentation( "IsMonoidOfUp2DimensionalMappingsObj",
    IsMonoidOfUp2DimensionalMappings and IsAttributeStoringRep,
    [ "Object2d", "ImagesList" ] );

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#A  ImagesList                                returns list of DerivationImages
#A  AllOrRegular                                    type of derivations record
#A  ImagesTable                                   returns lists of image lists
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DeclareAttribute( "ImagesList", IsMonoidOfUp2DimensionalMappings );
DeclareAttribute( "AllOrRegular", IsMonoidOfUp2DimensionalMappings );
DeclareAttribute( "ImagesTable", IsMonoidOfUp2DimensionalMappings );

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#O  MonoidOfUp2DimensionalMappingsObj( <obj>, <images>, <str> )
#F  MonoidOfUp2DimensionalMappingsFamily . . .  family for up-mappings monoid 
#T  MonoidOfUp2DimensionalMappingsType . . . . .  type for up-mappings monoid 
#P  IsMonoidOfDerivations                               
#P  IsMonoidOfSections
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DeclareOperation( "MonoidOfUp2DimensionalMappingsObj",
    [ Is2DimensionalDomain, IsHomogeneousList, IsString ] );
MonoidOfUp2DimensionalMappingsFamily := 
    CollectionsFamily( Up2DimensionalMappingFamily ); 
BindGlobal( "MonoidOfUp2DimensionalMappingsType", 
            NewType( MonoidOfUp2DimensionalMappingsFamily, 
                     IsMonoidOfUp2DimensionalMappingsObj ) ); 
DeclareProperty( "IsMonoidOfDerivations", IsMonoidOfUp2DimensionalMappings );
DeclareProperty( "IsMonoidOfSections", IsMonoidOfUp2DimensionalMappings );

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#A  RegularDerivations    find all invertible derivations for a crossed module
#A  AllDerivations                   find all derivations for a crossed module
#A  RegularSections              find all invertible sections for a cat1-group
#A  AllSections                             find all sections for a cat1-group
##
DeclareAttribute( "RegularDerivations", IsXMod );
DeclareAttribute( "AllDerivations", IsXMod );
DeclareAttribute( "RegularSections", IsCat1Group );
DeclareAttribute( "AllSections", IsCat1Group );

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#O  BacktrackDerivationsJ          recursive function for BacktrackDerivations
#O  BacktrackDerivations            recursive construction for all derivations
#O  BacktrackSectionsJ         recursion used by RegularSections & AllSections
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DeclareOperation( "BacktrackDerivationsJ",  [ IsXMod, IsHomogeneousList,
    IsHomogeneousList, IsHomogeneousList, IsInt, IsString ] );
DeclareOperation( "BacktrackDerivations", [ IsXMod, IsString ] );
DeclareOperation( "BacktrackSectionsJ", [ IsRecord, IsInt, IsObject, IsInt ] );

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#A  WhiteheadMonoidTable( XM )               Table of products of derivations
#A  WhiteheadGroupTable( XM )        Table of products of regular derivations
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##  ?? should these refer just to 2DimensionalDomains ??
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DeclareAttribute( "WhiteheadMonoidTable", IsXMod );
DeclareAttribute( "WhiteheadGroupTable", IsXMod );

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#A  WhiteheadPermGroup( XM )                     a permutation representation
#A  WhiteheadGroupGeneratingDerivations                   generators for W(X)
#A  WhiteheadGroupGeneratorPositions         positions of generators for W(X)
#A  WhiteheadTransMonoid( XM )                a transformation representation
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DeclareAttribute( "WhiteheadPermGroup", IsXMod );
DeclareAttribute( "WhiteheadGroupGeneratingDerivations", IsXMod );
DeclareAttribute( "WhiteheadGroupGeneratorPositions", IsXMod );
DeclareAttribute( "WhiteheadTransMonoid", IsXMod );

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#E  gp2up.gd  . . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here