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#############################################################################
##
#W mapprep.gi GAP library Thomas Breuer
#W & Martin Schönert
#W & Frank Celler
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains (representation dependent)
##
## 1. methods for general mappings in 'IsDefaultMappingRep'
## 2. methods for composition mappings,
## 3. methods for mappings by function,
## 4. methods for inverse mappings
## 5. methods for identity mappings
## 6. methods for zero mappings
##
#############################################################################
##
## 1. methods for general mappings in 'IsDefaultMappingRep'
##
#############################################################################
##
#R IsDefaultGeneralMappingRep( <map> )
##
## Source and range of such a general mapping are stored in its type, as
## 'DataType( TypeObj( <map> ) )[1]' resp.
## 'DataType( TypeObj( <map> ) )[2]'.
##
## Note that this representation does *not* decide whether <map> is
## a positional or a component object.
##
DeclareRepresentation( "IsDefaultGeneralMappingRep",
IsGeneralMapping and HasSource and HasRange,
[] );
#T methods to handle attributes 'One', 'Inverse', and 'InverseGeneralMapping',
#T 'ImagesSource', 'PreImagesRange'?
#############################################################################
##
#F TypeOfDefaultGeneralMapping( <source>, <range>, <filter> )
##
InstallGlobalFunction( TypeOfDefaultGeneralMapping,
function( source, range, filter )
local Type;
# Do a cheap test whether the general mapping has equal source and range.
if IsIdenticalObj( source, range ) then
filter:= filter and IsEndoGeneralMapping;
fi;
# Construct the type.
Type:= NewType( GeneralMappingsFamily(
ElementsFamily( FamilyObj( source ) ),
ElementsFamily( FamilyObj( range ) ) ),
IsDefaultGeneralMappingRep and filter );
# Store source and range.
SetDataType( Type, [ source, range ] );
# Return the type.
return Type;
end );
#############################################################################
##
#M Range( <map> )
##
InstallMethod( Range,
"for default general mapping",
true,
[ IsGeneralMapping and IsDefaultGeneralMappingRep ],
GETTER_FLAGS + 1, # higher than the system getter!
map -> DataType( TypeObj( map ) )[2] );
#############################################################################
##
#M Source( <map> )
##
InstallMethod( Source,
"for default general mapping",
true,
[ IsGeneralMapping and IsDefaultGeneralMappingRep ],
GETTER_FLAGS + 1, # higher than the system getter!
map -> DataType( TypeObj( map ) )[1] );
#############################################################################
##
## 2. methods for composition mappings,
##
#############################################################################
##
#F ConstituentsCompositionMapping( <map> )
##
InstallGlobalFunction(ConstituentsCompositionMapping,function(map)
if not IsCompositionMappingRep(map) then
Error("<map> must be `IsCompositionMappingRep'");
fi;
return [map!.map1,map!.map2];
end);
#############################################################################
##
#M CompositionMapping2( <map2>, <map1> ) . . . . . for two general mappings
##
InstallGlobalFunction(CompositionMapping2General,
function( map2, map1 )
local com; # composition of <map1> and <map2>, result
# Make the general mapping.
if IsSPGeneralMapping( map1 ) and IsSPGeneralMapping( map2 ) then
com:= Objectify( TypeOfDefaultGeneralMapping( Source( map1 ),
Range( map2 ),
IsCompositionMappingRep and IsSPGeneralMapping ),
rec() );
else
com:= Objectify( TypeOfDefaultGeneralMapping( Source( map1 ),
Range( map2 ),
IsCompositionMappingRep and IsNonSPGeneralMapping ),
rec() );
fi;
# Enter the identifying information.
# (Maintenance of useful information is dealt with by the
# wrapper function `CompositionMapping'.)
com!.map1:= map1;
com!.map2:= map2;
# Return the composition.
return com;
end );
InstallMethod( CompositionMapping2,
"for two general mappings",
FamSource1EqFamRange2,
[ IsGeneralMapping, IsGeneralMapping ], 0,
CompositionMapping2General);
#############################################################################
##
#M IsInjective( <map> ) . . . . . . . . . . . . . . for composition mapping
##
InstallMethod( IsInjective,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 0,
function( com )
if IsInjective( com!.map1 ) and IsInjective( com!.map2 ) then
return true;
fi;
if not IsInjective( com!.map1 ) and IsTotal( com!.map2 ) then
return false;
fi;
if IsSurjective( com!.map1 ) and IsSingleValued( com!.map1 )
and not IsInjective( com!.map2 ) then
return false;
fi;
TryNextMethod();
end );
#############################################################################
##
#M IsSingleValued( <map> ) . . . . . . . . . . . . for composition mapping
##
InstallMethod( IsSingleValued,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 0,
function( com )
if IsSingleValued( com!.map1 ) and IsSingleValued( com!.map2 ) then
return true;
fi;
if not IsSingleValued( com!.map1 )
and IsInjective( com!.map2 ) and IsTotal( com!.map2 ) then
return false;
fi;
if IsSurjective( com!.map1 ) and not IsSingleValued( com!.map2 ) then
return false;
fi;
TryNextMethod();
end );
#############################################################################
##
#M IsSurjective( <map> ) . . . . . . . . . . . . . . for composition mapping
##
InstallMethod( IsSurjective,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 0,
function( com )
if not IsSurjective( com!.map2 ) then
return false;
elif IsSurjective( com!.map1 ) and
IsSubset( Range( com!.map1 ), PreImagesRange( com!.map2 ) ) then
return true;
fi;
TryNextMethod();
end );
#############################################################################
##
#M IsTotal( <map> ) . . . . . . . . . . . . . . . . for composition mapping
##
InstallMethod( IsTotal,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 0,
function( com )
if not IsTotal( com!.map1 ) then
return false;
elif IsTotal( com!.map2 ) and
IsSubset( Source( com!.map2 ), ImagesSource( com!.map1 ) ) then
return true;
fi;
TryNextMethod();
end );
#############################################################################
##
#M ImagesElm( <map>, <elm> ) . . . . . . . . . . . . for composition mapping
##
InstallMethod( ImagesElm,
"for a composition mapping, and an element",
FamSourceEqFamElm,
[ IsCompositionMappingRep, IsObject ], 0,
function( com, elm )
local im;
im:= ImagesElm( com!.map1, elm );
if not IsEmpty( im ) then
return ImagesSet( com!.map2, im );
else
return [];
fi;
end );
#############################################################################
##
#M ImagesSet( <map>, <elms> ) . . . . . . . . . . . for composition mapping
##
InstallMethod( ImagesSet,
"for a composition mapping, and a collection",
CollFamSourceEqFamElms,
[ IsCompositionMappingRep, IsCollection ], 0,
function ( com, elms )
local im;
im:= ImagesSet( com!.map1, elms );
if not IsEmpty( im ) then
return ImagesSet( com!.map2, im );
else
return [];
fi;
end );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . . . . . . for composition mapping
##
InstallMethod( ImagesRepresentative,
"for a composition mapping, and an element",
FamSourceEqFamElm,
[ IsCompositionMappingRep, IsObject ], 0,
function( com, elm )
local im, rep;
im:= ImagesRepresentative( com!.map1, elm );
if im = fail then
# 'elm' has no images under 'com!.map1', so it has none under 'com'.
return fail;
else
im:= ImagesRepresentative( com!.map2, im );
if im <> fail then
return im;
fi;
# It may happen that only the chosen representative has no images.
for im in Enumerator( ImagesElm( com!.map1, elm ) ) do
rep:= ImagesRepresentative( com!.map2, im );
if rep <> fail then
return rep;
fi;
od;
return fail;
fi;
end );
#############################################################################
##
#M PreImagesElm( <map>, <elm> ) . . . . . . . . . . for composition mapping
##
InstallMethod( PreImagesElm,
"for a composition mapping, and an element",
FamRangeEqFamElm,
[ IsCompositionMappingRep, IsObject ], 0,
function( com, elm )
local im;
im:= PreImagesElm( com!.map2, elm );
if not IsEmpty( im ) then
return PreImagesSet( com!.map1, im );
else
return [];
fi;
end );
#############################################################################
##
#M PreImagesSet( <map>, <elm> ) . . . . . . . . . . for composition mapping
##
InstallMethod( PreImagesSet,
"for a composition mapping, and a collection",
CollFamRangeEqFamElms,
[ IsCompositionMappingRep, IsCollection ], 0,
function( com, elms )
local im;
im:= PreImagesSet( com!.map2, elms );
if not IsEmpty( im ) then
return PreImagesSet( com!.map1, im );
else
return [];
fi;
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . . . . for composition mapping
##
InstallMethod( PreImagesRepresentative,
"for a composition mapping, and an element",
FamRangeEqFamElm,
[ IsCompositionMappingRep, IsObject ], 0,
function( com, elm )
local im, rep;
im:= PreImagesRepresentative( com!.map2, elm );
if im = fail then
# 'elm' has no preimages under 'com!.map2', so it has none under 'com'.
return fail;
else
im:= PreImagesRepresentative( com!.map1, im );
if im <> fail then
return im;
fi;
# It may happen that only the chosen representative has no preimages.
for im in Enumerator( PreImagesElm( com!.map2, elm ) ) do
rep:= PreImagesRepresentative( com!.map1, im );
if rep <> fail then
return rep;
fi;
od;
return fail;
fi;
end );
#############################################################################
##
#M KernelOfAdditiveGeneralMapping( <map> ) . . . . . for composition mapping
##
InstallMethod( KernelOfAdditiveGeneralMapping,
"for a composition mapping that resp. add. and add.inv.",
true,
[ IsGeneralMapping and IsCompositionMappingRep
and RespectsAddition and RespectsAdditiveInverses ], 0,
function( com )
if IsInjective( com!.map2 ) then
return KernelOfAdditiveGeneralMapping( com!.map1 );
else
return PreImagesSet( com!.map1,
KernelOfAdditiveGeneralMapping( com!.map2 ) );
fi;
end );
#############################################################################
##
#M CoKernelOfAdditiveGeneralMapping( <map> ) . . . . for composition mapping
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
"for a composition mapping that resp. add. and add.inv.",
true,
[ IsGeneralMapping and IsCompositionMappingRep
and RespectsAddition and RespectsAdditiveInverses ], 0,
function( com )
if IsSingleValued( com!.map1 ) then
return CoKernelOfAdditiveGeneralMapping( com!.map2 );
else
return ImagesSet( com!.map2,
CoKernelOfAdditiveGeneralMapping( com!.map1 ) );
fi;
end );
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <map> ) . . for composition mapping
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,
"for a composition mapping that resp. mult. and inv.",
true,
[ IsGeneralMapping and IsCompositionMappingRep
and RespectsMultiplication and RespectsInverses ], 0,
function( com )
if IsInjective( com!.map2 ) then
return KernelOfMultiplicativeGeneralMapping( com!.map1 );
else
return PreImagesSet( com!.map1,
KernelOfMultiplicativeGeneralMapping( com!.map2 ) );
fi;
end );
#############################################################################
##
#M CoKernelOfMultiplicativeGeneralMapping( <map> ) . for composition mapping
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping,
"for a composition mapping that resp. mult. and inv.",
true,
[ IsGeneralMapping and IsCompositionMappingRep
and RespectsMultiplication and RespectsInverses ], 0,
function( com )
if IsSingleValued( com!.map1 ) then
return CoKernelOfMultiplicativeGeneralMapping( com!.map2 );
else
return ImagesSet( com!.map2,
CoKernelOfMultiplicativeGeneralMapping( com!.map1 ) );
fi;
end );
#############################################################################
##
#M ViewObj( <map> ) . . . . . . . . . . . . . . . . for composition mapping
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . for composition mapping
##
InstallMethod( ViewObj,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 100,
function( com )
Print( "CompositionMapping( ", BHINT );
View( com!.map2 );
Print( ",", BHINT, " " );
View( com!.map1 );
Print( " )", BHINT );
end );
InstallMethod( PrintObj,
"for a composition mapping",
true,
[ IsCompositionMappingRep ], 100,
function( com )
Print( "CompositionMapping( ", com!.map2, ", ", com!.map1, " )" );
end );
#############################################################################
##
## 3. methods for mappings by function,
##
#############################################################################
##
#R IsMappingByFunctionRep( <map> )
##
DeclareRepresentation( "IsMappingByFunctionRep",
IsMapping and IsAttributeStoringRep,
[ "fun" ] );
#T really attribute storing ??
#############################################################################
##
#R IsMappingByFunctionWithInverseRep( <map> )
##
DeclareRepresentation( "IsMappingByFunctionWithInverseRep",
IsMappingByFunctionRep
and IsBijective,
#T 1996/10/10 fceller where to put non-reps, 4th position?
[ "fun", "invFun" ] );
#############################################################################
##
#R IsNonSPMappingByFunctionRep( <map> )
##
DeclareRepresentation( "IsNonSPMappingByFunctionRep",
IsNonSPGeneralMapping and IsMappingByFunctionRep, [] );
#############################################################################
##
#R IsNonSPMappingByFunctionWithInverseRep( <map> )
##
DeclareRepresentation( "IsNonSPMappingByFunctionWithInverseRep",
IsMappingByFunctionWithInverseRep and IsNonSPMappingByFunctionRep,
[ "fun", "invFun" ] );
#############################################################################
##
#R IsSPMappingByFunctionRep( <map> )
##
DeclareRepresentation( "IsSPMappingByFunctionRep",
IsSPGeneralMapping and IsMappingByFunctionRep, [] );
#############################################################################
##
#R IsSPMappingByFunctionWithInverseRep( <map> )
##
DeclareRepresentation( "IsSPMappingByFunctionWithInverseRep",
IsMappingByFunctionWithInverseRep and IsSPMappingByFunctionRep,
[ "fun", "invFun" ] );
#############################################################################
##
#F MappingByFunction( <D>, <E>, <fun> ) . . . . . create map from function
#F MappingByFunction( <D>, <E>, <fun>, <invfun> )
##
InstallGlobalFunction( MappingByFunction, function ( arg )
local map; # mapping <map>, result
if not Length(arg) in [3,4,5] then
# signal an error
Error( "usage: MappingByFunction( <D>, <E>, <fun>[, <inv>] )" );
fi;
# ensure that the source and range are domains
if not (IsDomain(arg[1]) and IsDomain(arg[2])) then
Error("MappingByFunction: Source and Range must be domains");
fi;
# no inverse function given
if Length(arg)<>4 then
# make the general mapping
map:= Objectify( TypeOfDefaultGeneralMapping( arg[1], arg[2],
IsNonSPMappingByFunctionRep
and IsSingleValued
and IsTotal ),
rec( fun:= arg[3] ) );
if Length(arg)=5 and IsFunction(arg[5]) then
map!.prefun:=arg[5];
fi;
# inverse function given
elif Length(arg) = 4 then
# make the mapping
map:= Objectify( TypeOfDefaultGeneralMapping( arg[1], arg[2],
IsNonSPMappingByFunctionWithInverseRep
and IsBijective ),
rec( fun := arg[3],
invFun := arg[4],
prefun := arg[4]) );
fi;
# return the mapping
return map;
end );
#############################################################################
##
#M ImageElm( <map>, <elm> ) . . . . . . . . . . . . for mapping by function
##
InstallMethod( ImageElm,
"for mapping by function",
FamSourceEqFamElm,
[ IsMappingByFunctionRep, IsObject ], 0,
function ( map, elm )
return map!.fun( elm );
end );
#############################################################################
##
#M ImagesElm( <map>, <elm> ) . . . . . . . . . . . . for mapping by function
##
InstallMethod( ImagesElm,
"for mapping by function",
FamSourceEqFamElm,
[ IsMappingByFunctionRep, IsObject ], 0,
function ( map, elm )
return [ map!.fun( elm ) ];
end );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . . . . . . for mapping by function
##
InstallMethod( ImagesRepresentative,
"for mapping by function",
FamSourceEqFamElm,
[ IsMappingByFunctionRep, IsObject ], 0,
function ( map, elm )
return map!.fun( elm );
end );
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <map> ) . for mapping by function
##
InstallMethod(KernelOfMultiplicativeGeneralMapping,"hom by function",true,
[ IsMappingByFunctionRep and IsGroupHomomorphism ],0,
function ( map )
return
KernelOfMultiplicativeGeneralMapping(AsGroupGeneralMappingByImages(map));
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . . . . for mapping by function
##
InstallMethod( PreImagesRepresentative,
"for mapping by function",
FamRangeEqFamElm,
[ IsMappingByFunctionRep, IsObject ], 0,
function ( map, elm )
if not IsBound(map!.prefun) then
# no quick way known
TryNextMethod();
fi;
return map!.prefun( elm );
end );
#############################################################################
##
#M PreImageElm( <map>, <elm> ) . . . . . . . . . . . for mapping by function
##
InstallMethod( PreImageElm,
"for mapping by function",
FamRangeEqFamElm,
[ IsMappingByFunctionWithInverseRep, IsObject ], 0,
function ( map, elm )
return map!.invFun( elm );
end );
#############################################################################
##
#M PreImagesElm( <map>, <elm> ) . . . . . . . . . . for mapping by function
##
InstallMethod( PreImagesElm,
"for mapping by function",
FamRangeEqFamElm,
[ IsMappingByFunctionWithInverseRep, IsObject ], 0,
function ( map, elm )
return [ map!.invFun( elm ) ];
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . . . . for mapping by function
##
InstallMethod( PreImagesRepresentative,
"for mapping by function with inverse",
FamRangeEqFamElm,
[ IsMappingByFunctionWithInverseRep, IsObject ], 0,
function ( map, elm )
return map!.invFun( elm );
end );
#############################################################################
##
## Transfer information about the mapping map to its associated inverse
## mapping inv. For example, if map is total than inv is surjective and
## vice versa.
##
BindGlobal( "TransferMappingPropertiesToInverse", function ( map, inv )
# If possible, enter preimage and image.
if HasImagesSource( map ) then
SetPreImagesRange( inv, ImagesSource( map ) );
fi;
if HasPreImagesRange( map ) then
SetImagesSource( inv, PreImagesRange( map ) );
fi;
# Maintain important properties.
if HasIsSingleValued( map ) then
SetIsInjective( inv, IsSingleValued( map ) );
fi;
if HasIsInjective( map ) then
SetIsSingleValued( inv, IsInjective( map ) );
fi;
if HasIsTotal( map ) then
SetIsSurjective( inv, IsTotal( map ) );
fi;
if HasIsSurjective( map ) then
SetIsTotal( inv, IsSurjective( map ) );
fi;
if HasIsEndoGeneralMapping( map ) then
SetIsEndoGeneralMapping( inv, IsEndoGeneralMapping( map ) );
fi;
# Maintain the maintainings w.r.t. multiplication.
if HasRespectsMultiplication( map ) then
SetRespectsMultiplication( inv, RespectsMultiplication( map ) );
fi;
if HasRespectsInverses( map ) then
SetRespectsInverses( inv, RespectsInverses( map ) );
elif HasRespectsOne( map ) then
SetRespectsOne( inv, RespectsOne( map ) );
fi;
# Maintain the maintainings w.r.t. addition.
if HasRespectsAddition( map ) then
SetRespectsAddition( inv, RespectsAddition( map ) );
fi;
if HasRespectsAdditiveInverses( map ) then
SetRespectsAdditiveInverses( inv, RespectsAdditiveInverses( map ) );
elif HasRespectsZero( map ) then
SetRespectsZero( inv, RespectsZero( map ) );
fi;
# Maintain respecting of scalar multiplication.
# (Note the slight asymmetry, depending on the coefficient domains.)
if HasRespectsScalarMultiplication( map )
and LeftActingDomain( Source( map ) )
= LeftActingDomain( Range( map ) ) then
SetRespectsScalarMultiplication( inv,
RespectsScalarMultiplication( map ) );
fi;
# We know the inverse general mapping of the inverse general mapping ;-).
SetInverseGeneralMapping( inv, map );
end );
#############################################################################
##
#M InverseGeneralMapping( <map> ) . . . . . . . . . for mapping by function
##
InstallMethod( InverseGeneralMapping, "for mapping by function", true,
[ IsMappingByFunctionWithInverseRep ], 0,
function ( map )
local inv;
inv:= MappingByFunction( Range( map ), Source( map ),
map!.invFun, map!.fun );
TransferMappingPropertiesToInverse( map, inv );
return inv;
end );
InstallMethod( RestrictedInverseGeneralMapping, "for mapping by function", true,
[ IsMappingByFunctionWithInverseRep ], 0,
function ( map )
local inv;
inv:= MappingByFunction( Image( map ), Source( map ),
map!.invFun, map!.fun );
TransferMappingPropertiesToInverse( map, inv );
return inv;
end );
#############################################################################
##
#M ViewObj( <map> ) . . . . . . . . . . . . . . . . for mapping by function
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . for mapping by function
##
InstallMethod( ViewObj,
"for mapping by function",
true,
[ IsMappingByFunctionRep ], 0,
function ( map )
Print( "MappingByFunction( " );
View( Source( map ) );
Print( ", " );
View( Range( map ) );
Print( ", " );
View( map!.fun );
Print( " )" );
end );
InstallMethod( PrintObj,
"for mapping by function",
true,
[ IsMappingByFunctionRep ], 0,
function ( map )
Print( "MappingByFunction( ",
Source( map ), ", ", Range( map ), ", ",
map!.fun, " )" );
end );
#############################################################################
##
#M ViewObj( <map> ) . . . . . . . . . for mapping by function with inverse
#M PrintObj( <map> ) . . . . . . . . . for mapping by function with inverse
##
InstallMethod( ViewObj,
"for mapping by function with inverse",
true,
[ IsMappingByFunctionWithInverseRep ], 0,
function ( map )
Print( "MappingByFunction( " );
View( Source( map ) );
Print( ", " );
View( Range( map ) );
Print( ", " );
View( map!.fun );
Print( ", " );
View( map!.invFun );
Print( " )" );
end );
InstallMethod( PrintObj,
"for mapping by function with inverse",
true,
[ IsMappingByFunctionWithInverseRep ], 0,
function ( map )
Print( "MappingByFunction( ",
Source( map ), ", ", Range( map ), ", ",
map!.fun, ", ", map!.invFun, " )" );
end );
#############################################################################
##
## 4. methods for inverse mappings
##
#############################################################################
##
#R IsInverseGeneralMappingRep( <map> )
##
## Note that if a mapping knows its inverse mapping then also the inverse
## mapping knows its inverse mapping.
## So we need this flag to avoid infinite recursion when a question is
## delegated to the inverse of a mapping.
##
DeclareRepresentation( "IsInverseGeneralMappingRep",
IsNonSPGeneralMapping,
[] );
#############################################################################
##
#M InverseGeneralMapping( <map> ) . for a general mapping with known inverse
##
InstallImmediateMethod( InverseGeneralMapping,
IsGeneralMapping and HasInverse and IsAttributeStoringRep, 0,
function( map )
if Inverse( map ) <> fail then
return Inverse( map );
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M InverseGeneralMapping( <map> ) . . . . . . . . . . for a general mapping
##
InstallMethod( InverseGeneralMapping,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
function ( map )
local inv;
# Make the mapping.
inv:= Objectify( TypeOfDefaultGeneralMapping( Range( map ),
Source( map ),
IsInverseGeneralMappingRep
and IsAttributeStoringRep ),
rec() );
TransferMappingPropertiesToInverse( map, inv );
# Return the inverse general mapping.
return inv;
end );
#############################################################################
##
#M IsSingleValued( <map> ) . . . . . . . . . . . . . for an inverse mapping
##
InstallMethod( IsSingleValued,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> IsInjective( InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M IsInjective( <map> ) . . . . . . . . . . . . . . for an inverse mapping
##
InstallMethod( IsInjective,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> IsSingleValued( InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M IsSurjective( <map> ) . . . . . . . . . . . . . . for an inverse mapping
##
InstallMethod( IsSurjective,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> IsTotal( InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M IsTotal( <map> ) . . . . . . . . . . . . . . . . for an inverse mapping
##
InstallMethod( IsTotal,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> IsSurjective( InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M CoKernelOfAdditiveGeneralMapping( <invmap> ) . . . . for inverse mapping
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> KernelOfAdditiveGeneralMapping(
InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M KernelOfAdditiveGeneralMapping( <invmap> ) . . . . . for inverse mapping
##
InstallMethod( KernelOfAdditiveGeneralMapping,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> CoKernelOfAdditiveGeneralMapping(
InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M CoKernelOfMultiplicativeGeneralMapping( <invmap> ) . for inverse mapping
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> KernelOfMultiplicativeGeneralMapping(
InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <invmap> ) . . for inverse mapping
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 0,
inv -> CoKernelOfMultiplicativeGeneralMapping(
InverseGeneralMapping( inv ) ) );
#############################################################################
##
#M ImageElm( <invmap>, <map> ) . . . . . . . for inverse mapping and element
##
InstallMethod( ImageElm,
"for an inverse mapping and an element",
FamSourceEqFamElm,
[ IsMapping and IsInverseGeneralMappingRep, IsObject ], 0,
function ( inv, elm )
return PreImageElm( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M ImagesElm( <invmap>, <map> ) . . . . . . for inverse mapping and element
##
InstallMethod( ImagesElm,
"for an inverse mapping and an element",
FamSourceEqFamElm,
[ IsGeneralMapping and IsInverseGeneralMappingRep, IsObject ], 0,
function ( inv, elm )
return PreImagesElm( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M ImagesSet( <invmap>, <coll> ) . . . . for inverse mapping and collection
##
InstallMethod( ImagesSet,
"for an inverse mapping and a collection",
CollFamSourceEqFamElms,
[ IsGeneralMapping and IsInverseGeneralMappingRep, IsCollection ], 0,
function ( inv, elms )
return PreImagesSet( InverseGeneralMapping( inv ), elms );
end );
#############################################################################
##
#M ImagesSet( <invmap>, <coll> ) . . . . for inverse mapping and collection
##
InstallMethod( ImagesRepresentative,
"for an inverse mapping and an element",
FamSourceEqFamElm,
[ IsGeneralMapping and IsInverseGeneralMappingRep, IsObject ], 0,
function ( inv, elm )
return PreImagesRepresentative( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M PreImageElm( <invmap>, <elm> ) . . . . . for inverse mapping and element
##
InstallMethod( PreImageElm,
"for an inj. & surj. inverse mapping, and an element",
FamRangeEqFamElm,
[ IsGeneralMapping and IsInverseGeneralMappingRep
and IsInjective and IsSurjective, IsObject ], 0,
function ( inv, elm )
return ImageElm( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M PreImagesElm( <invmap>, <elm> ) . . . . . for inverse mapping and element
##
InstallMethod( PreImagesElm,
"for an inverse mapping and an element",
FamRangeEqFamElm,
[ IsGeneralMapping and IsInverseGeneralMappingRep, IsObject ], 0,
function ( inv, elm )
return ImagesElm( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M PreImagesSet( <invmap>, <coll> ) . . for inverse mapping and collection
##
InstallMethod( PreImagesSet,
"for an inverse mapping and a collection",
CollFamRangeEqFamElms,
[ IsGeneralMapping and IsInverseGeneralMappingRep, IsCollection ], 0,
function ( inv, elms )
return ImagesSet( InverseGeneralMapping( inv ), elms );
end );
#############################################################################
##
#M PreImagesRepresentative( <invmap>, <elm> ) . . for inv. mapping and elm.
##
InstallMethod( PreImagesRepresentative,
"for an inverse mapping and an element",
FamRangeEqFamElm,
[ IsInverseGeneralMappingRep, IsObject ], 0,
function ( inv, elm )
return ImagesRepresentative( InverseGeneralMapping( inv ), elm );
end );
#############################################################################
##
#M ViewObj( <invmap> ) . . . . . . . . . . . . . . . . . . for inv. mapping
#M PrintObj( <invmap> ) . . . . . . . . . . . . . . . . . for inv. mapping
##
InstallMethod( ViewObj,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 100,
function ( inv )
Print( "InverseGeneralMapping( " );
View( InverseGeneralMapping( inv ) );
Print( " )" );
end );
InstallMethod( PrintObj,
"for an inverse mapping",
true,
[ IsGeneralMapping and IsInverseGeneralMappingRep ], 100,
function ( inv )
Print( "InverseGeneralMapping( ", InverseGeneralMapping( inv )," )" );
end );
#############################################################################
##
## 5. methods for identity mappings
##
## For each domain we need to construct only one identity mapping.
## In order to allow this to interact with other mappings of this domain
## (for example, with automorphisms of a field in a special representation),
## one needs to install methods to compare these mappings with the identity
## mapping via '\=' and '\<'.
##
## Methods for identity mappings are all installed with rank 'SUM_FLAGS'.
##
#############################################################################
##
## An identity mapping whose source has a nice structure gets the properties
## to respect this structure.
##
BindGlobal( "ImmediateImplicationsIdentityMapping", function( idmap )
local source;
source:= Source( idmap );
# multiplicative structure
if IsMagma( source ) then
SetRespectsMultiplication( idmap, true );
if IsMagmaWithOne( source ) then
SetRespectsOne( idmap, true );
if IsMagmaWithInverses( source ) then
SetRespectsInverses( idmap, true );
fi;
fi;
fi;
# additive structure
if IsAdditiveMagma( source ) then
SetRespectsAddition( idmap, true );
if IsAdditiveMagmaWithZero( source ) then
SetRespectsZero( idmap, true );
if IsAdditiveGroup( source ) then
SetRespectsAdditiveInverses( idmap, true );
# linear structure
if IsLeftModule( source ) then
SetRespectsScalarMultiplication( idmap, true );
fi;
fi;
fi;
fi;
end );
#############################################################################
##
#M IdentityMapping( <D> ) . . . . . . . . identity mapping of a collection
##
InstallMethod( IdentityMapping,
"for a collection",
true,
[ IsCollection ], 0,
function( D )
local id;
# make the mapping
id := Objectify( TypeOfDefaultGeneralMapping( D, D,
IsSPGeneralMapping
and IsAdditiveElementWithInverse
and IsAttributeStoringRep
and IsOne ),
rec() );
# the identity mapping is self-inverse
SetInverseGeneralMapping( id, id );
# set the respectings
ImmediateImplicationsIdentityMapping( id );
# return the identity mapping
return id;
end );
#############################################################################
##
#M \^( <idmap>, <n> ) . . . . . . . . . . for identity mapping and integer
##
InstallMethod( \^,
"for identity mapping and integer",
true,
[ IsGeneralMapping and IsOne, IsInt ],
SUM_FLAGS, # can't do better
function ( id, n )
return id;
end );
#############################################################################
##
#M ImageElm( <idmap>, <elm> ) . . . . . . for identity mapping and element
##
InstallMethod( ImageElm,
"for identity mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return elm;
end );
#############################################################################
##
#M ImagesElm( <idmap>, <elm> ) . . . . . . for identity mapping and element
##
InstallMethod( ImagesElm,
"for identity mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return [ elm ];
end );
#############################################################################
##
#M ImagesSet( <idmap>, <coll> ) . . . . for identity mapping and collection
##
InstallMethod( ImagesSet,
"for identity mapping and collection",
CollFamSourceEqFamElms,
[ IsGeneralMapping and IsOne, IsCollection ],
SUM_FLAGS, # can't do better
function ( id, elms )
return elms;
end );
#############################################################################
##
#M ImagesSource( <idmap> )
##
InstallMethod( ImagesSource,"for identity mapping",true,
[ IsGeneralMapping and IsOne ], SUM_FLAGS, # can't do better
function ( id )
return Source(id);
end );
#############################################################################
##
#M ImagesRepresentative( <idmap>, <elm> ) for identity mapping and element
##
InstallMethod( ImagesRepresentative,
"for identity mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return elm;
end );
#############################################################################
##
#M PreImageElm( <idmap>, <elm> ) . . . . for identity mapping and element
##
InstallMethod( PreImageElm,
"for identity mapping and object",
FamRangeEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return elm;
end );
#############################################################################
##
#M PreImagesElm( <idmap>, <elm> ) . . . . for identity mapping and element
##
InstallMethod( PreImagesElm,
"for identity mapping and object",
FamRangeEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return [ elm ];
end );
#############################################################################
##
#M PreImagesSet( <idmap>, <coll> ) . . . for identity mapping and collection
##
InstallMethod( PreImagesSet,
"for identity mapping and collection",
CollFamRangeEqFamElms,
[ IsGeneralMapping and IsOne, IsCollection ],
SUM_FLAGS, # can't do better
function ( id, elms )
return elms;
end );
#############################################################################
##
#M PreImagesRepresentative( <idmap>, <elm> )
##
InstallMethod( PreImagesRepresentative,
"for identity mapping and object",
FamRangeEqFamElm,
[ IsGeneralMapping and IsOne, IsObject ],
SUM_FLAGS, # can't do better
function ( id, elm )
return elm;
end );
#############################################################################
##
#M ViewObj( <idmap> ) . . . . . . . . . . . . . . . . for identity mapping
#M PrintObj( <idmap> ) . . . . . . . . . . . . . . . . for identity mapping
##
InstallMethod( ViewObj,
"for identity mapping",
true,
[ IsGeneralMapping and IsOne ], SUM_FLAGS,
function ( id )
Print( "IdentityMapping( " );
View( Source( id ) );
Print( " )" );
end );
InstallMethod( PrintObj,
"for identity mapping",
true,
[ IsGeneralMapping and IsOne ], SUM_FLAGS,
function ( id )
Print( "IdentityMapping( ", Source( id ), " )" );
end );
#############################################################################
##
#M CompositionMapping2( <map>, <idmap> ) . for gen. mapping and id. mapping
##
InstallMethod( CompositionMapping2,
"for general mapping and identity mapping", FamSource1EqFamRange2,
[ IsGeneralMapping, IsGeneralMapping and IsOne ],
SUM_FLAGS + 1, # should be higher than the rank for a zero mapping
function ( map, id )
if not IsSubset(Range(id),Source(map)) then
# if the identity is defined on something smaller, we need to take a
# true `CompositionMapping'.
TryNextMethod();
fi;
return map;
end );
#############################################################################
##
#M CompositionMapping2( <idmap>, <map> ) . for id. mapping and gen. mapping
##
InstallMethod( CompositionMapping2,
"for identity mapping and general mapping",FamSource1EqFamRange2,
[ IsGeneralMapping and IsOne, IsGeneralMapping ],
SUM_FLAGS + 1, # should be higher than the rank for a zero mapping
function( id, map )
if not IsSubset(Source(id),Range(map)) then
# if the identity is defined on something smaller, we need to take a
# true `CompositionMapping'.
TryNextMethod();
fi;
return map;
end );
#############################################################################
##
## 6. methods for zero mappings
##
## methods for zero mappings are all installed with rank 'SUM_FLAGS'
##
#T (use 'IsZero' in '\+' method for mappings ...)
#############################################################################
##
## A zero mapping whose source has a nice structure gets the properties
## to respect this structure.
##
BindGlobal( "ImmediateImplicationsZeroMapping", function( zeromap )
local source;
source:= Source( zeromap );
# multiplicative structure
if IsMagma( source ) then
SetRespectsMultiplication( zeromap, true );
if IsMagmaWithOne( source ) then
SetRespectsOne( zeromap, false );
if IsMagmaWithInverses( source ) then
SetRespectsInverses( zeromap, false );
fi;
fi;
fi;
# additive structure
if IsAdditiveMagma( source ) then
SetRespectsAddition( zeromap, true );
if IsAdditiveMagmaWithZero( source ) then
SetRespectsZero( zeromap, true );
if IsAdditiveGroup( source ) then
SetRespectsAdditiveInverses( zeromap, true );
fi;
fi;
fi;
# linear structure
if IsLeftModule( source ) then
SetRespectsScalarMultiplication( zeromap, true );
fi;
end );
#############################################################################
##
#F ZeroMapping( <source>, <range> )
##
## maps every element of <source> to 'Zero( <range> )'.
## This is independent of the structure of <source> and <range>.
##
InstallMethod( ZeroMapping,
"for collection and additive-magma-with-zero",
true,
[ IsCollection, IsAdditiveMagmaWithZero ], 0,
function( S, R )
local zero; # the zero mapping, result
# make the mapping
zero := Objectify( TypeOfDefaultGeneralMapping( S, R,
IsSPGeneralMapping
and IsAdditiveElementWithInverse
and IsAttributeStoringRep
and IsZero ),
rec() );
# set the respectings
ImmediateImplicationsZeroMapping( zero );
# return the zero mapping
return zero;
end );
#############################################################################
##
#M \^( <zeromap>, <n> ) . . . . . . . for zero mapping and positive integer
##
InstallMethod( \^,
"for zero mapping and positive integer",
true,
[ IsGeneralMapping and IsZero, IsPosInt ], SUM_FLAGS,
function( zero, n )
if Zero( Source( zero ) ) in Range( zero ) then
return zero;
else
Error( "source and range of <zero> do not match" );
fi;
end );
#############################################################################
##
#M ImagesSource( <zeromap> ) . . . . . . . . . . . . . . . for zero mapping
##
InstallMethod( ImagesSource,
"for zero mapping",
true,
[ IsGeneralMapping and IsZero ], SUM_FLAGS,
function( zero )
if IsAdditiveMagmaWithZero( Range( zero ) ) then
return TrivialSubadditiveMagmaWithZero( Range( zero ) );
else
return [ Zero( Range( zero ) ) ];
fi;
end );
#############################################################################
##
#M ImageElm( <zeromap>, <elm> ) . . . . . . . for zero mapping and element
##
InstallMethod( ImageElm,
"for zero mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsZero, IsObject ], SUM_FLAGS,
function( zero, elm )
return Zero( Range( zero ) );
end );
#############################################################################
##
#M ImagesElm( <zeromap>, <elm> ) . . . . . . . for zero mapping and element
##
InstallMethod( ImagesElm,
"for zero mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsZero, IsObject ], SUM_FLAGS,
function( zero, elm )
return [ Zero( Range( zero ) ) ];
end );
#############################################################################
##
#M ImagesSet( <zeromap>, <coll> ) . . . . . for zero mapping and collection
##
InstallMethod( ImagesSet,
"for zero mapping and collection",
CollFamSourceEqFamElms,
[ IsGeneralMapping and IsZero, IsCollection ], SUM_FLAGS,
function( zero, elms )
return TrivialSubadditiveMagmaWithZero( Range( zero ) );
end );
#############################################################################
##
#M ImagesRepresentative( <zeromap>, <elm> ) . for zero mapping and element
##
InstallMethod( ImagesRepresentative,
"for zero mapping and object",
FamSourceEqFamElm,
[ IsGeneralMapping and IsZero, IsObject ], SUM_FLAGS,
function( zero, elm )
return Zero( Range( zero ) );
end );
#############################################################################
##
#M PreImagesElm( <zeromap>, <elm> ) . . . . . for zero mapping and element
##
InstallMethod( PreImagesElm,
"for zero mapping and object",
FamRangeEqFamElm,
[ IsGeneralMapping and IsZero, IsObject ], SUM_FLAGS,
function( zero, elm )
if elm = Zero( Range( zero ) ) then
return Source( zero );
else
return [];
fi;
end );
#############################################################################
##
#M PreImagesSet( <zeromap>, <elms> ) . . . . for zero mapping and collection
##
InstallMethod( PreImagesSet,
"for zero mapping and collection",
CollFamRangeEqFamElms,
[ IsGeneralMapping and IsZero, IsCollection ], SUM_FLAGS,
function( zero, elms )
if Zero( Range( zero ) ) in elms then
return Source( zero );
else
return [];
fi;
end );
#############################################################################
##
#M PreImagesRepresentative( <zeromap>, <elm> )
##
InstallMethod( PreImagesRepresentative,
"for zero mapping and object",
FamRangeEqFamElm,
[ IsGeneralMapping and IsZero, IsObject ], SUM_FLAGS,
function( zero, elm )
if elm = Zero( Range( zero ) ) then
return Zero( Source( zero ) );
else
return fail;
fi;
end );
#############################################################################
##
#M ViewObj( <zeromap> ) . . . . . . . . . . . . . . . . . for zero mapping
#M PrintObj( <zeromap> ) . . . . . . . . . . . . . . . . . for zero mapping
##
InstallMethod( ViewObj,
"for zero mapping",
true,
[ IsGeneralMapping and IsZero ], SUM_FLAGS,
function( zero )
Print( "ZeroMapping( " );
View( Source( zero ) );
Print( ", " );
View( Range( zero ) );
Print( " )" );
end );
InstallMethod( PrintObj,
"for zero mapping",
true,
[ IsGeneralMapping and IsZero ], SUM_FLAGS,
function( zero )
Print( "ZeroMapping( ", Source( zero ), ", ", Range( zero ), " )" );
end );
#############################################################################
##
#M CompositionMapping2( <map>, <zeromap> ) for gen. mapping and zero mapping
##
InstallMethod( CompositionMapping2,
"for general mapping and zero mapping",
FamSource1EqFamRange2,
[ IsGeneralMapping, IsGeneralMapping and IsZero ], SUM_FLAGS,
function( map, zero )
return ZeroMapping( Source( map ), Range( zero ) );
end );
#############################################################################
##
#M CompositionMapping2( <zeromap>, <map> ) for zero mapping and gen. mapping
##
InstallMethod( CompositionMapping2,
"for zero mapping and single-valued gen. mapping that resp. zero",
FamSource1EqFamRange2,
[ IsGeneralMapping and IsZero,
IsGeneralMapping and IsSingleValued and RespectsZero ],
SUM_FLAGS,
function( zero, map )
return ZeroMapping( Source( zero ), Range( map ) );
end );
#############################################################################
##
#M IsInjective( <zeromap> ) . . . . . . . . . . . . . . . for zero mapping
##
InstallMethod( IsInjective,
"for zero mapping",
true,
[ IsGeneralMapping and IsZero ], 0,
zero -> Size( Source( zero ) ) = 1 );
#############################################################################
##
#M IsSurjective( <zeromap> ) . . . . . . . . . . . . . . . for zero mapping
##
InstallMethod( IsSurjective,
"for zero mapping",
true,
[ IsGeneralMapping and IsZero ], 0,
zero -> Size( Range( zero ) ) = 1 );
#############################################################################
##
## 7. methods for general restricted mappings,
##
#############################################################################
##
#M GeneralRestrictedMapping( <map>, <source>,<range> )
##
InstallGlobalFunction(GeneralRestrictedMapping,
function( map, s,r )
local res, prop;
# Make the general mapping.
if IsSPGeneralMapping( map ) then
res:= Objectify( TypeOfDefaultGeneralMapping( s,r,
IsGeneralRestrictedMappingRep and IsSPGeneralMapping ),
rec() );
else
res:= Objectify( TypeOfDefaultGeneralMapping( s,r,
IsGeneralRestrictedMappingRep and IsNonSPGeneralMapping ),
rec() );
fi;
# Enter the identifying information.
res!.map:= map;
SetSource(res,s);
SetRange(res,r);
for prop in [IsSingleValued, IsTotal, IsInjective, RespectsMultiplication, , RespectsInverses,
RespectsAddition, RespectsAdditiveInverses, RespectsScalarMultiplication] do
if Tester(prop)(map) and prop(map) then
Setter(prop)(res, true);
fi;
od;
# Return the restriction.
return res;
end );
#############################################################################
##
#M ImagesElm( <map>, <elm> ) . . . . . . . . . . . . for restricted mapping
##
InstallMethod( ImagesElm,
"for a restricted mapping, and an element",
FamSourceEqFamElm,
[ IsGeneralRestrictedMappingRep, IsObject ], 0,
function( res, elm )
local im;
im:= ImagesElm( res!.map, elm );
if not ( (HasIsSingleValued(res) and IsSingleValued(res)) or
(HasIsSingleValued(res!.map) and IsSingleValued(res!.map)) ) then
im:=Intersection(Range(res),im);
fi;
return im;
end );
#############################################################################
##
#M ImagesSet( <map>, <elms> ) . . . . . . . . . . . for restricted mapping
##
InstallMethod( ImagesSet,
"for a restricted mapping, and a collection",
CollFamSourceEqFamElms,
[ IsGeneralRestrictedMappingRep, IsCollection ], 0,
function ( res, elms )
local im;
im:= ImagesSet( res!.map, elms );
if not ( (HasIsSingleValued(res) and IsSingleValued(res)) or
(HasIsSingleValued(res!.map) and IsSingleValued(res!.map)) ) then
im:=Intersection(Range(res),im);
fi;
return im;
end );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . . . . . . for restricted mapping
##
InstallMethod( ImagesRepresentative,
"for a restricted mapping, and an element",
FamSourceEqFamElm,
[ IsGeneralRestrictedMappingRep, IsObject ], 0,
function( res, elm )
local im;
im:= ImagesRepresentative( res!.map, elm );
if im = fail then
# 'elm' has no images under 'res!.map', so it has none under 'res'.
return fail;
elif im in Range(res) then
return im;
elif HasIsSingleValued(res!.map) and IsSingleValued(res!.map) then
return fail; # no other choice
else
# It may happen that only the chosen representative is not in im
im:= ImagesElm( res!.map, elm );
return First(im,i->i in Range(res));
fi;
end );
#############################################################################
##
#M PreImagesElm( <map>, <elm> ) . . . . . . . . . . for restricted mapping
##
InstallMethod( PreImagesElm,
"for a restricted mapping, and an element",
FamRangeEqFamElm,
[ IsGeneralRestrictedMappingRep, IsObject ], 0,
function( res, elm )
local preim;
preim:= PreImagesElm( res!.map, elm );
if not ( (HasIsInjective(res) and IsInjective(res)) or
(HasIsInjective(res!.map) and IsInjective(res!.map)) ) then
preim:=Intersection(Source(res),preim);
fi;
return preim;
end );
#############################################################################
##
#M PreImagesSet( <map>, <elm> ) . . . . . . . . . . for restricted mapping
##
InstallMethod( PreImagesSet,
"for a restricted mapping, and a collection",
CollFamRangeEqFamElms,
[ IsGeneralRestrictedMappingRep, IsCollection ], 0,
function( res, elms )
local preim;
preim:= PreImagesSet( res!.map, elms );
if not ( (HasIsInjective(res) and IsInjective(res)) or
(HasIsInjective(res!.map) and IsInjective(res!.map)) ) then
preim:=Intersection(Source(res),preim);
fi;
return preim;
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . . . . for restricted mapping
##
InstallMethod( PreImagesRepresentative,
"for a restricted mapping, and an element",
FamRangeEqFamElm,
[ IsGeneralRestrictedMappingRep, IsObject ], 0,
function( res, elm )
local preim, rep;
preim:= PreImagesRepresentative( res!.map, elm );
if preim = fail then
# 'elm' has no preimages under 'res!.map', so it has none under 'res'.
return fail;
elif preim in Source(res) then
return preim;
elif HasIsInjective(res!.map) and IsInjective(res!.map) then
return fail; # no other choice
else
preim:= PreImages( res!.map, elm );
return First(preim,x->x in Source(res));
fi;
end );
#############################################################################
##
#M KernelOfAdditiveGeneralMapping( <map> ) . . . . . for restricted mapping
##
InstallMethod( KernelOfAdditiveGeneralMapping,
"for a restricted mapping that resp. add. and add.inv.", true,
[ IsGeneralMapping and IsGeneralRestrictedMappingRep
and RespectsAddition and RespectsAdditiveInverses ], 0,
function( res )
return Intersection(Source(res),KernelOfAdditiveGeneralMapping( res!.map ));
end );
#############################################################################
##
#M CoKernelOfAdditiveGeneralMapping( <map> ) . . . . for restricted mapping
##
InstallMethod( CoKernelOfAdditiveGeneralMapping,
"for a restricted mapping that resp. add. and add.inv.",
true,
[ IsGeneralMapping and IsGeneralRestrictedMappingRep
and RespectsAddition and RespectsAdditiveInverses ], 0,
function( res )
return Intersection(Range(res),CoKernelOfAdditiveGeneralMapping( res!.map ));
end );
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <map> ) . . for restricted mapping
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,
"for a restricted mapping that resp. mult. and inv.",
true,
[ IsGeneralMapping and IsGeneralRestrictedMappingRep
and RespectsMultiplication and RespectsInverses ], 0,
function( res )
return Intersection(Source(res),
KernelOfMultiplicativeGeneralMapping(res!.map));
end );
#############################################################################
##
#M CoKernelOfMultiplicativeGeneralMapping( <map> ) . for restricted mapping
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping,
"for a restricted mapping that resp. mult. and inv.",
true,
[ IsGeneralMapping and IsGeneralRestrictedMappingRep
and RespectsMultiplication and RespectsInverses ], 0,
function( res )
return Intersection(Range(res),
CoKernelOfMultiplicativeGeneralMapping(res!.map));
end );
#############################################################################
##
#M ViewObj( <map> ) . . . . . . . . . . . . . . . . for restricted mapping
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . for restricted mapping
##
InstallMethod( ViewObj,
"for a restricted mapping",
true,
[ IsGeneralRestrictedMappingRep ], 100,
function( res )
Print( "GeneralRestrictedMapping( " );
View( res!.map );
Print( ", " );
View( Source(res) );
Print( ", " );
View( Range(res) );
Print( " )" );
end );
InstallMethod( PrintObj,
"for a restricted mapping",
true,
[ IsGeneralRestrictedMappingRep ], 100,
function( res )
Print( "GeneralRestrictedMapping( ", res!.map, ", ", Source(res),
",", Range(res)," )" );
end );
#############################################################################
##
#M RestrictedMapping(<hom>,<U>)
##
InstallMethod(RestrictedMapping,"for mapping that is already restricted",
CollFamSourceEqFamElms,
[IsGeneralMapping and IsGeneralRestrictedMappingRep, IsDomain],
SUM_FLAGS,
function(hom, U)
return GeneralRestrictedMapping (hom!.map, U, Range(hom!.map));
end);
#############################################################################
##
#M RestrictedMapping(<hom>,<U>)
##
InstallMethod(RestrictedMapping,"use GeneralRestrictedMapping",
CollFamSourceEqFamElms,[IsGeneralMapping,IsDomain],0,
function(hom, U)
return GeneralRestrictedMapping (hom, U, Range(hom));
end);
#############################################################################
##
#E