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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Martin Schönert, Frank Celler.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains
## 1. the design of families of general mappings
## now moved to mapping1.gi since it contain GAP/HPCGAP divergence
## 2. generic methods for general mappings
## 3. generic methods for underlying relations of general mappings
##
#############################################################################
##
## 2. generic methods for general mappings
##
#############################################################################
##
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . . . for general mapping
##
InstallMethod( PrintObj,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
function( map )
Print( "<general mapping: ", Source( map ), " -> ", Range( map ), " >" );
end );
#############################################################################
##
#M PrintObj( <map> ) . . . . . . . . . . . . . . . . . . . . . . for mapping
##
InstallMethod( PrintObj,
"for a mapping",
true,
[ IsMapping ], 0,
function( map )
Print( "<mapping: ", Source( map ), " -> ", Range( map ), " >" );
end );
#T these are `ViewObj' methods. How could real `PrintObj' methods look like?
# #############################################################################
# ##
# #M IsOne( <map> ) . . . . . . . . . . . . . . . . . . . for general mapping
# ##
# InstallOtherMethod( IsOne,
# "for general mapping",
# true,
# [ IsGeneralMapping ], 0,
# map -> Source( map ) = Range( map )
# and IsBijective( map )
# and ForAll( Source( map ), elm -> ImageElm( map, elm ) = elm ) );
#############################################################################
##
#M IsZero( <map> ) . . . . . . . . . . . . . . . . . . . for general mapping
##
InstallOtherMethod( IsZero,
"for general mapping",
true,
[ IsGeneralMapping ], 0,
map -> Zero( Range( map ) ) <> fail
and IsTotal( map )
and ImagesSource( map ) = [ Zero( Range( map ) ) ] );
#############################################################################
##
#M IsEndoGeneralMapping( <map> ) . . . . . . . . . . . . for general mapping
##
InstallOtherMethod( IsEndoGeneralMapping,
"for general mapping",
true,
[ IsGeneralMapping ], 0,
map -> Source( map ) = Range( map ) );
#############################################################################
##
#F Image( <map> ) . . . . set of images of the source of a general mapping
#F Image( <map>, <elm> ) . . . . unique image of an element under a mapping
#F Image( <map>, <coll> ) . . set of images of a collection under a mapping
##
InstallGlobalFunction( Image, function ( arg )
local map, # mapping <map>, first argument
elm; # element <elm>, second argument
if Length(arg) > 0 and not IsGeneralMapping( arg[1] ) then
ErrorNoReturn( "<map> must be a general mapping" );
fi;
# image of the source under <map>, which may be multi valued in this case
if Length( arg ) = 1 then
return ImagesSource( arg[1] );
elif Length( arg ) = 2 then
map := arg[1];
elm := arg[2];
# image of a single element <elm> under the mapping <map>
if FamSourceEqFamElm( FamilyObj( map ), FamilyObj( elm ) ) then
if not IsMapping( map ) then
ErrorNoReturn( "<map> must be single-valued and total" );
elif not elm in Source( map ) then
ErrorNoReturn( "<elm> must be an element of Source(<map>)" );
fi;
return ImageElm( map, elm );
# image of a collection of elements <elm> under the mapping <map>
elif CollFamSourceEqFamElms( FamilyObj( map ), FamilyObj(elm) ) then
if not IsSubset( Source( map ), elm ) then
ErrorNoReturn( "the collection <elm> must be contained in ",
"Source(<map>)" );
fi;
if IsDomain( elm ) or IsSSortedList( elm ) then
if HasSource(map) and IsIdenticalObj(Source(map),elm) then
return ImagesSource( map );
else
return ImagesSet( map, elm );
fi;
elif IsHomogeneousList( elm ) then
return ImagesSet( map, Set( elm ) );
fi;
# image of the empty list
elif IsList( elm ) and IsEmpty( elm ) then
return [];
else
ErrorNoReturn( "the families of the element or collection <elm> ",
"and Source(<map>) don't match, ",
"maybe <elm> is not contained in Source(<map>) or ",
"is not a homogeneous list or collection" );
fi;
fi;
ErrorNoReturn( "usage: Image(<map>), Image(<map>,<elm>), ",
"Image(<map>,<coll>)" );
end );
#############################################################################
##
#M <map>(<elm>)
#M <map>(<coll>)
##
## Method to allow mappings to be used like functions when appropriate
InstallMethod(CallFuncList, [IsGeneralMapping, IsList],
function(map, lst)
if Length(lst) <> 1 then
TryNextMethod();
fi;
return Image(map, lst[1]);
end);
#############################################################################
##
#F Images( <map> ) . . . . set of images of the source of a general mapping
#F Images( <map>, <elm> ) . . . set of images of an element under a mapping
#F Images( <map>, <coll> ) . . set of images of a collection under a mapping
##
InstallGlobalFunction( Images, function ( arg )
local map, # mapping <map>, first argument
elm; # element <elm>, second argument
if Length(arg) > 0 and not IsGeneralMapping( arg[1] ) then
ErrorNoReturn( "<map> must be a general mapping" );
fi;
# image of the source under <map>
if Length( arg ) = 1 then
return ImagesSource( arg[1] );
elif Length( arg ) = 2 then
map := arg[1];
elm := arg[2];
# image of a single element <elm> under the mapping <map>
if FamSourceEqFamElm( FamilyObj( map ), FamilyObj( elm ) ) then
if not elm in Source( map ) then
ErrorNoReturn( "<elm> must be an element of Source(<map>)" );
fi;
return ImagesElm( map, elm );
# image of a collection of elements <elm> under the mapping <map>
elif CollFamSourceEqFamElms( FamilyObj( map ), FamilyObj(elm) ) then
if not IsSubset( Source( map ), elm ) then
ErrorNoReturn( "the collection <elm> must be contained in ",
"Source(<map>)" );
fi;
if IsDomain( elm ) or IsSSortedList( elm ) then
return ImagesSet( map, elm );
elif IsHomogeneousList( elm ) then
return ImagesSet( map, Set( elm ) );
fi;
# image of the empty list
elif IsList( elm ) and IsEmpty( elm ) then
return [];
else
ErrorNoReturn( "the families of the element or collection <elm> ",
"and Source(<map>) don't match, ",
"maybe <elm> is not contained in Source(<map>) or ",
"is not a homogeneous list or collection" );
fi;
fi;
ErrorNoReturn( "usage: Images(<map>), Images(<map>,<elm>), ",
"Images(<map>,<coll>)" );
end );
#############################################################################
##
#F PreImage( <map> ) . . set of preimages of the range of a general mapping
#F PreImage( <map>, <elm> ) . unique preimage of an elm under a gen.mapping
#F PreImage(<map>,<coll>) set of preimages of a coll. under a gen. mapping
##
InstallGlobalFunction( PreImage, function ( arg )
local map, # gen. mapping <map>, first argument
img; # element <img>, second argument
if Length( arg ) > 0 and not IsGeneralMapping( arg[1] ) then
ErrorNoReturn( "<map> must be a general mapping" );
fi;
# preimage of the range under <map>, which may be a general mapping
if Length( arg ) = 1 then
return PreImagesRange( arg[1] );
elif Length( arg ) = 2 then
map := arg[1];
img := arg[2];
# preimage of a single element <img> under <map>
if FamRangeEqFamElm( FamilyObj( map ), FamilyObj( img ) ) then
if not ( IsInjective( map ) and IsSurjective( map ) ) then
ErrorNoReturn( "<map> must be an injective and surjective ",
"mapping" );
elif not img in Range( map ) then
ErrorNoReturn( "<elm> must be an element of Range(<map>)" );
fi;
return PreImageElm( map, img );
# preimage of a collection of elements <img> under <map>
elif CollFamRangeEqFamElms( FamilyObj( map ), FamilyObj( img ) ) then
if not IsSubset( Range( map ), img ) then
ErrorNoReturn( "the collection <elm> must be contained in ",
"Range(<map>)" );
fi;
if IsDomain( img ) or IsSSortedList( img ) then
return PreImagesSet( map, img );
elif IsHomogeneousList( img ) then
return PreImagesSet( map, Set( img ) );
fi;
# preimage of the empty list
elif IsList( img ) and IsEmpty( img ) then
return [];
else
ErrorNoReturn( "the families of the element or collection <elm> ",
"and Range(<map>) don't match, ",
"maybe <elm> is not contained in Range(<map>) or ",
"is not a homogeneous list or collection" );
fi;
fi;
ErrorNoReturn( "usage: PreImage(<map>), PreImage(<map>,<img>), ",
"PreImage(<map>,<coll>)" );
end );
#############################################################################
##
#F PreImages( <map> ) . . . set of preimages of the range of a gen. mapping
#F PreImages(<map>,<elm>) . set of preimages of an elm under a gen. mapping
#F PreImages(<map>,<coll>) set of preimages of a coll. under a gen. mapping
##
InstallGlobalFunction( PreImages, function ( arg )
local map, # mapping <map>, first argument
img; # element <img>, second argument
# preimage of the range under <map>
if Length( arg ) = 1 then
return PreImagesRange( arg[1] );
elif Length( arg ) = 2 then
map := arg[1];
img := arg[2];
if not IsGeneralMapping( map ) then
ErrorNoReturn( "<map> must be a general mapping" );
fi;
# preimage of a single element <img> under <map>
if FamRangeEqFamElm( FamilyObj( map ), FamilyObj( img ) ) then
if not img in Range( map ) then
ErrorNoReturn( "<elm> must be an element of Range(<map>)" );
fi;
return PreImagesElm( map, img );
# preimage of a collection of elements <img> under <map>
elif CollFamRangeEqFamElms( FamilyObj( map ), FamilyObj( img ) ) then
if not IsSubset( Range( map ), img ) then
ErrorNoReturn( "the collection <elm> must be contained in ",
"Range(<map>)" );
fi;
if IsDomain( img ) or IsSSortedList( img ) then
return PreImagesSet( map, img );
elif IsHomogeneousList( img ) then
return PreImagesSet( map, Set( img ) );
fi;
# preimage of the empty list
elif IsList( img ) and IsEmpty( img ) then
return [];
else
ErrorNoReturn( "the families of the element or collection <elm> ",
"and Range(<map>) don't match, ",
"maybe <elm> is not contained in Range(<map>) or ",
"is not a homogeneous list or collection" );
fi;
fi;
ErrorNoReturn( "usage: PreImages(<map>), PreImages(<map>,<img>), ",
"PreImages(<map>,<coll>)" );
end );
#############################################################################
##
#F CompositionMapping(<map1>,<map2>, ... ) . . . . . composition of mappings
##
InstallGlobalFunction( CompositionMapping, function ( arg )
local com, # composition of the arguments, result
nxt, # next general mapping in the composition
new, # intermediate composition
i; # loop variable
# check the arguments
if Length( arg ) = 0 then
Error("usage: CompositionMapping(<map1>..)");
fi;
# unravel the argument list
if Length( arg ) = 1 and IsList( arg[1] ) then
arg := arg[1];
fi;
# compute the composition
com := Last( arg );
if not IsGeneralMapping( com ) then
Error( "<com> must be (general) mapping" );
fi;
for i in Reversed( [1..Length( arg )-1] ) do
nxt:= arg[i];
# Check that the composition can be formed.
if not IsGeneralMapping( nxt ) then
Error( "<i>-th argument must be (general) mapping" );
elif not FamSource2EqFamRange1( FamilyObj( com ),
FamilyObj( nxt ) ) then
Error( "the range of <com> and the source of <nxt> ",
"must be contained in the same family" );
fi;
# Compute the composition.
new := CompositionMapping2( nxt, com );
# Maintain properties
# (Do only *cheap* tests, otherwise one could attempt to check
# `IsSubset( Source( nxt ), ImagesSource( com ) )' in the case
# of `IsTotal', and to check in the case of `IsSurjective' whether
# `IsSubset( Range( com ), PreImagesRange( nxt ) )' holds.)
if HasIsSingleValued( com ) and IsSingleValued( com )
and HasIsSingleValued( nxt ) and IsSingleValued( nxt ) then
SetIsSingleValued( new, true );
fi;
if HasIsInjective( com ) and IsInjective( com )
and HasIsInjective( nxt ) and IsInjective( nxt ) then
SetIsInjective( new, true );
fi;
if HasIsTotal( com ) and IsTotal( com )
and HasIsTotal( nxt ) and IsTotal( nxt )
and ((HasImagesSource(com) and
CanComputeIsSubset(Source(nxt),ImagesSource(com)) and
IsSubset(Source(nxt),ImagesSource(com)))
or (HasRange(com) and
CanComputeIsSubset(Source(nxt),Range(com)) and
IsSubset(Source(nxt),Range(com))) ) then
SetIsTotal( new, true );
fi;
if HasRange(com) and IsIdenticalObj(Source(nxt),Range(com)) then
if HasIsTotal( com ) and IsTotal( com )
and HasIsTotal( nxt ) and IsTotal( nxt ) then
SetIsTotal( new, true );
fi;
if HasIsSurjective( com ) and IsSurjective( com )
and HasIsSurjective( nxt ) and IsSurjective( nxt ) then
SetIsSurjective( new, true );
fi;
fi;
# Maintain respectings.
if HasRespectsAddition( com )
and HasRespectsAddition( nxt )
and RespectsAddition( com )
and RespectsAddition( nxt ) then
SetRespectsAddition( new, true );
fi;
if HasRespectsAdditiveInverses( com )
and HasRespectsAdditiveInverses( nxt )
and RespectsAdditiveInverses( com )
and RespectsAdditiveInverses( nxt ) then
SetRespectsAdditiveInverses( new, true );
elif HasRespectsZero( com )
and HasRespectsZero( nxt )
and RespectsZero( com )
and RespectsZero( nxt ) then
SetRespectsZero( new, true );
fi;
if HasRespectsMultiplication( com )
and HasRespectsMultiplication( nxt )
and RespectsMultiplication( com )
and RespectsMultiplication( nxt ) then
SetRespectsMultiplication( new, true );
fi;
if HasRespectsInverses( com )
and HasRespectsInverses( nxt )
and RespectsInverses( com )
and RespectsInverses( nxt ) then
SetRespectsInverses( new, true );
elif HasRespectsOne( com )
and HasRespectsOne( nxt )
and RespectsOne( com )
and RespectsOne( nxt ) then
SetRespectsOne( new, true );
fi;
if IsIdenticalObj( Source( nxt ), Range( com ) )
and HasRespectsScalarMultiplication( com )
and HasRespectsScalarMultiplication( nxt )
and RespectsScalarMultiplication( com )
and RespectsScalarMultiplication( nxt ) then
# Note that equality of the two relevant domains
# does in general not suffice to get linearity,
# since their left acting domains must fit, too.
SetRespectsScalarMultiplication( new, true );
fi;
com:= new;
od;
if IsIdenticalObj( Source( com ), Range( com ) ) then
SetIsEndoGeneralMapping( com, true );
fi;
# Return the composition.
return com;
end );
# Temporarily disabled -- See #569
# Currently some group homomrophisms construct inverse maps that are really
# restricted inverses (i.e. defined only on the image). Together with these
# immediate methods this can cause wrong indications of IsSurjective etc.
# for these maps. While this needs to be fixed in the future properly, the
# immediate methods are temporarily disabled to avoid this error having
# further effects.
# #############################################################################
# ##
# #M IsInjective( <map> ) . . . . . for gen. mapp. with known inv. gen. mapp.
# #M IsSingleValued( <map> ) . . . . for gen. mapp. with known inv. gen. mapp.
# #M IsSurjective( <map> ) . . . . . for gen. mapp. with known inv. gen. mapp.
# #M IsTotal( <map> ) . . . . . . . for gen. mapp. with known inv. gen. mapp.
# ##
# InstallImmediateMethod( IsInjective,
# IsGeneralMapping and HasInverseGeneralMapping, 0,
# function( map )
# map:= InverseGeneralMapping( map );
# if HasIsSingleValued( map ) then
# return IsSingleValued( map );
# else
# TryNextMethod();
# fi;
# end );
#
# InstallImmediateMethod( IsSingleValued,
# IsGeneralMapping and HasInverseGeneralMapping, 0,
# function( map )
# map:= InverseGeneralMapping( map );
# if HasIsInjective( map ) then
# return IsInjective( map );
# else
# TryNextMethod();
# fi;
# end );
#
# InstallImmediateMethod( IsSurjective,
# IsGeneralMapping and HasInverseGeneralMapping, 0,
# function( map )
# map:= InverseGeneralMapping( map );
# if HasIsTotal( map ) then
# return IsTotal( map );
# else
# TryNextMethod();
# fi;
# end );
#
# InstallImmediateMethod( IsTotal,
# IsGeneralMapping and HasInverseGeneralMapping, 0,
# function( map )
# map:= InverseGeneralMapping( map );
# if HasIsSurjective( map ) then
# return IsSurjective( map );
# else
# TryNextMethod();
# fi;
# end );
#############################################################################
##
#M IsTotal( <map> ) . . . . . . . . . . . . . . . . . . for general mapping
##
InstallMethod( IsTotal, "for a general mapping", true,
[ IsGeneralMapping ], 0,
function( map )
# For a total and injective general mapping,
# the range cannot be smaller than the source.
if HasIsInjective( map ) and IsInjective( map )
and CanComputeSize(Range(map)) and CanComputeSize(Source(map))
and Size( Range( map ) ) < Size( Source( map ) ) then
return false;
else
return IsSubset( PreImagesRange( map ), Source( map ) );
fi;
end );
#############################################################################
##
#M IsSurjective( <map> ) . . . . . . . . . . . . . . . . for general mapping
##
InstallMethod( IsSurjective, "for a general mapping", true,
[ IsGeneralMapping ], 0,
function( map )
# For a single-valued and surjective general mapping,
# the source cannot be smaller than the range.
if HasIsSingleValued( map ) and IsSingleValued( map )
and CanComputeSize(Range(map)) and CanComputeSize(Source(map))
and Size( Source( map ) ) < Size( Range( map ) ) then
return false;
else
return IsSubset( ImagesSource( map ), Range( map ) );
fi;
end );
#############################################################################
##
#M IsSingleValued( <map> ) . . . . . . . . . . . . . . for a general mapping
##
InstallMethod( IsSingleValued, "for a general mapping", true,
[ IsGeneralMapping ], 0,
function( map )
if HasIsSurjective( map ) and IsSurjective( map )
and CanComputeSize(Range(map)) and CanComputeSize(Source(map)) then
# For a single-valued and surjective general mapping,
# the range cannot be larger than the source.
if Size( Source( map ) ) < Size( Range( map ) ) then
return false;
fi;
fi;
if IsFinite( Source( map ) ) then
# test that each element of the source has at most one image
return ForAll( Source( map ),
elm -> Size( ImagesElm( map, elm ) ) <= 1 );
else
# give up if <map> has an infinite source
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsInjective( <map> ) . . . . . . . . . . . . . . . for a general mapping
##
InstallMethod( IsInjective, "for a general mapping", true,
[ IsGeneralMapping ], 0,
function( map )
local enum, # enumerator for the source
imgs, # list of images for the elements of the source
elm, # loop over `enum'
img; # one set of images
if HasIsTotal( map ) and IsTotal( map ) then
# For a total and injective general mapping,
# the source cannot be larger than the range.
if Size( Range( map ) ) < Size( Source( map ) )
and CanComputeSize(Range(map)) and CanComputeSize(Source(map)) then
return false;
fi;
fi;
if IsFinite( Source( map ) ) then
# Check that the images of different elements are disjoint.
enum:= Enumerator( Source( map ) );
imgs:= [];
for elm in enum do
img:= ImagesElm( map, elm );
if ForAny( imgs, im -> Size( Intersection2( im, img ) ) <> 0 ) then
return false;
fi;
Add( imgs, img );
od;
return true;
else
# give up if <map> has an infinite source
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsInjective( <map> ) . . . . . . . . . . . . . . . . . . . for a mapping
##
InstallMethod( IsInjective, "for a mapping", true,
[ IsGeneralMapping and IsTotal and IsSingleValued ], 0,
function( map )
# For a total and injective general mapping,
# the source cannot be larger than the range.
if Size( Range( map ) ) < Size( Source( map ) )
and CanComputeSize(Range(map)) and CanComputeSize(Source(map)) then
return false;
# compare the size of the source with the size of the image
elif IsFinite( Source( map ) ) then
return Size( Source( map ) ) = Size( ImagesSource( map ) );
# give up if <map> has an infinite source
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M \=( <map1>, <map2> ) . . . . . . . . . . . . . for two general mappings
##
InstallMethod( \=,
"for two general mappings",
IsIdenticalObj,
[ IsGeneralMapping, IsGeneralMapping ], 0,
function( map1, map2 )
# Maybe the properties we already know determine the result.
if ( HasIsTotal( map1 ) and HasIsTotal( map2 )
and IsTotal( map1 ) <> IsTotal( map2 ) )
or ( HasIsSingleValued( map1 ) and HasIsSingleValued( map2 )
and IsSingleValued( map1 ) <> IsSingleValued( map2 ) )
or ( HasIsInjective( map1 ) and HasIsInjective( map2 )
and IsInjective( map1 ) <> IsInjective( map2 ) )
or ( HasIsSurjective( map1 ) and HasIsSurjective( map2 )
and IsSurjective( map1 ) <> IsSurjective( map2 ) )
then
return false;
fi;
# Otherwise we must really test the equality.
return Source( map1 ) = Source( map2 )
and Range( map1 ) = Range( map2 )
and UnderlyingRelation( map1 ) = UnderlyingRelation( map2 );
end );
#############################################################################
##
#M \<( <map1>, <map2> ) . . . . . . . . . . . . . for two general mappings
##
## Compare the sources, the ranges, the underlying relation.
##
InstallMethod( \<,
"for two general mappings",
IsIdenticalObj,
[ IsGeneralMapping, IsGeneralMapping ], 0,
function( map1, map2 )
if Source( map1 ) <> Source( map2 ) then
return Source( map1 ) < Source( map2 );
elif Range( map1 ) <> Range( map2 ) then
return Range( map1 ) < Range( map2 );
else
return UnderlyingRelation( map1 ) < UnderlyingRelation( map2 );
fi;
end );
#############################################################################
##
#M \+( <map>, <zero> ) . . . . . . . . for general mapping and zero mapping
##
InstallOtherMethod( \+,
"for general mapping and zero mapping",
IsIdenticalObj,
[ IsGeneralMapping, IsGeneralMapping and IsZero ], 0,
ReturnFirst );
#############################################################################
##
#M \+( <zero>, <map> ) . . . . . . . . for zero mapping and general mapping
##
InstallOtherMethod( \+,
"for zero mapping and general mapping",
IsIdenticalObj,
[ IsGeneralMapping and IsZero, IsGeneralMapping ], 0,
function( zero, map )
return map;
end );
#############################################################################
##
#M \*( <map1>, <map2> ) . . . . . . . . . . . . . for two general mappings
##
InstallMethod( \*, "for two general mappings", FamSource2EqFamRange1,
[ IsGeneralMapping, IsGeneralMapping ], 0,
function( map1, map2 )
return CompositionMapping( map2, map1 );
end );
#############################################################################
##
#M \^( <map1>, <map2> ) . . . . . . . . conjugation of two general mappings
##
InstallMethod( \^,
#T or shall this involve the usual inverse?
#T (then <map2> must be a bijection from its source to its source)
"for two general mappings",
FamSourceRgtEqFamsLft,
[ IsGeneralMapping, IsGeneralMapping ], 0,
function( lft, rgt )
return InverseGeneralMapping( rgt ) * lft * rgt;
end );
#############################################################################
##
#M \^( <elm>, <map> )
#T what about <coll> \^ <map> ?
##
InstallOtherMethod( \^,
"for element in the source, and general mapping",
FamElmEqFamSource,
[ IsObject, IsGeneralMapping ], 0,
function( elm, map )
return ImageElm( map, elm );
end );
#############################################################################
##
#M OneOp( <map> ) . . . . . . . . . . . . . . . . . . . . identity mapping
##
InstallOtherMethod( OneOp,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
function( map )
if IsEndoGeneralMapping( map ) then
return IdentityMapping( Source( map ) );
else
return fail;
fi;
end );
#############################################################################
##
#M ZeroOp( <map> ) . . . . . . . . . . . . . . . . . . . . . . zero mapping
##
InstallOtherMethod( ZeroOp,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
map -> ZeroMapping( Source( map ), Range( map ) ) );
#############################################################################
##
#M InverseOp( <map> ) . . . . . . . . . delegate to `InverseGeneralMapping'
##
InstallMethod( InverseOp,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
function( map )
local inv;
if IsEndoGeneralMapping( map ) and IsBijective( map ) then
inv := InverseGeneralMapping( map );
SetIsEndoGeneralMapping( inv, true );
SetIsBijective (inv, true); # this may seem superfluous, but
# IsInjective may create an InverseGeneralMapping which does not
# know that it is bijective
return inv;
else
Info(InfoWarning,1,
"The mapping must be bijective and have source=range\n",
"#I You might want to use `InverseGeneralMapping'");
return fail;
fi;
end );
#############################################################################
##
#M \*( <zero>, <map> ) . . . . . . . . . for zero and total general mapping
##
InstallMethod( \*,
"for zero and total general mapping",
FamElmEqFamRange,
[ IsRingElement and IsZero, IsGeneralMapping and IsTotal ], 0,
function( zero, map )
if IsGeneralMapping( zero ) then
TryNextMethod();
else
return ZeroMapping( Source( map ), Range( map ) );
fi;
end );
#############################################################################
##
#M <elm> / <map> . . . . . . . . . . . . . . . . . . . . preimage of element
##
InstallOtherMethod( \/,
"for element, and inj. & surj. general mapping",
FamElmEqFamRange,
[ IsObject, IsGeneralMapping and IsInjective and IsSurjective ], 0,
function( elm, map )
return PreImageElm( map, elm );
end );
#############################################################################
##
#M ImageElm( <map>, <elm> ) . . . . . . . . . . . . for mapping and element
##
InstallOtherMethod( ImageElm,
"for general mapping, and element",
FamSourceEqFamElm,
[ IsGeneralMapping, IsObject ], 0,
function( map, elm )
if not ( IsSingleValued( map ) and IsTotal( map ) ) then
Error( "<map> must be single-valued and total" );
fi;
return ImageElm( map, elm );
end );
InstallMethod( ImageElm,
"for mapping, and element",
FamSourceEqFamElm,
[ IsGeneralMapping and IsTotal and IsSingleValued, IsObject ], 0,
ImagesRepresentative );
#############################################################################
##
#M ImagesElm( <map>, <elm> ) . . . for non s.p. general mapping and element
##
InstallMethod( ImagesElm,
"for non s.p. general mapping, and element",
FamSourceEqFamElm,
[ IsNonSPGeneralMapping, IsObject ], 0,
function( map, elm )
Error( "no default function to compute images of <elm> under <map>" );
end );
#############################################################################
##
#M ImagesElm( <map>, <elm> ) . . for const. time access gen. map., and elm.
##
InstallMethod( ImagesElm,
"for constant time access general mapping, and element",
FamSourceEqFamElm,
[ IsGeneralMapping and IsConstantTimeAccessGeneralMapping, IsObject ], 0,
function( map, elm )
local imgs, pair;
imgs:= [];
for pair in GeneratorsOfDomain( UnderlyingRelation( map ) ) do
if pair[1] = elm then
AddSet( imgs, pair[2] );
fi;
od;
return imgs;
end );
#############################################################################
##
#M ImagesSet( <map>, <elms> ) . . for generel mapping and finite collection
##
InstallMethod( ImagesSet,
"for general mapping, and finite collection",
CollFamSourceEqFamElms,
[ IsGeneralMapping, IsCollection ], 0,
function( map, elms )
local imgs, elm;
if not IsFinite( elms ) then
TryNextMethod();
fi;
imgs:= [];
for elm in Enumerator( elms ) do
UniteSet( imgs, AsList( ImagesElm( map, elm ) ) );
od;
return imgs;
end );
InstallMethod( ImagesSet,
"for general mapping, and empty list",
true,
[ IsGeneralMapping, IsList and IsEmpty ], 0,
function( map, elms )
return [];
end );
#############################################################################
##
#M ImagesSource( <map> ) . . . . . . . . . . . . . . . . for general mapping
##
InstallMethod( ImagesSource,
"for general mapping",
true,
[ IsGeneralMapping ], 0,
map -> ImagesSet( map, Source( map ) ) );
#############################################################################
##
#M ImagesSource( <map> ) . . . . . . . . . . for surjective general mapping
##
InstallMethod( ImagesSource,
"for surjective general mapping (delegate to `Range')",
true,
[ IsGeneralMapping and IsSurjective ],
SUM_FLAGS, # immediately delegate, don;'t try anything else
Range );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . . . for s.p. gen. mapping and elm
##
InstallMethod( ImagesRepresentative,
"for s.p. general mapping, and element",
FamSourceEqFamElm,
[ IsSPGeneralMapping, IsObject ], 0,
function( map, elm )
Error( "no default method for s.p. general mapping" );
end );
#############################################################################
##
#M ImagesRepresentative( <map>, <elm> ) . for non s.p. gen. mapping and elm
##
InstallMethod( ImagesRepresentative,
"for non s.p. general mapping, and element",
FamSourceEqFamElm,
[ IsNonSPGeneralMapping, IsObject ], 0,
function( map, elm )
local
imgs; # all images of <elm> under <map>
# get all images of <elm> under <map>
imgs:= ImagesElm( map, elm );
# check that <elm> has at least one image under <map>
if IsEmpty( imgs ) then
return fail;
fi;
# pick one image, and return it
return Representative( imgs );
end );
#############################################################################
##
#M PreImageElm( <map>, <elm> )
##
InstallOtherMethod( PreImageElm,
"for general mapping, and element",
FamRangeEqFamElm,
[ IsGeneralMapping, IsObject ], 0,
function( map, elm )
if not ( IsInjective( map ) and IsSurjective( map ) ) then
Error( "<map> must be injective and surjective" );
fi;
return PreImageElm( map, elm );
end );
InstallMethod( PreImageElm,
"for inj. & surj. general mapping, and element",
FamRangeEqFamElm,
[ IsGeneralMapping and IsInjective and IsSurjective, IsObject ], 0,
PreImagesRepresentative );
#############################################################################
##
#M PreImagesElm( <map>, <elm> ) . . . . . . for general mapping and element
##
## more or less delegate to `ImagesElm'
##
InstallMethod( PreImagesElm,
"for general mapping with finite source, and element",
FamRangeEqFamElm,
[ IsGeneralMapping, IsObject ], 0,
function ( map, elm )
# for a finite source simply run over the elements of the source
if IsFinite( Source( map ) ) then
return Filtered( Source( map ),
pre -> elm in ImagesElm( map, pre ) );
# give up if <map> has an infinite source
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M PreImagesElm( <map>, <elm> ) for const. time access gen. map., and elm.
##
InstallMethod( PreImagesElm,
"for constant time access general mapping, and element",
FamRangeEqFamElm,
[ IsGeneralMapping and IsConstantTimeAccessGeneralMapping, IsObject ], 0,
function( map, elm )
local preimgs, pair;
preimgs:= [];
for pair in GeneratorsOfDomain( UnderlyingRelation( map ) ) do
if pair[2] = elm then
AddSet( preimgs, pair[1] );
fi;
od;
return preimgs;
end );
#############################################################################
##
#M PreImagesSet( <map>, <elms> ) . for general mapping and finite collection
##
InstallMethod( PreImagesSet,
"for general mapping, and finite collection",
CollFamRangeEqFamElms,
[ IsGeneralMapping, IsCollection ], 0,
function( map, elms )
local primgs, elm;
if not IsFinite( elms ) then
TryNextMethod();
fi;
primgs:= [];
for elm in Enumerator( elms ) do
UniteSet( primgs, AsList( PreImagesElm( map, elm ) ) );
od;
return primgs;
end );
InstallMethod( PreImagesSet,
"for general mapping, and empty list",
true,
[ IsGeneralMapping, IsList and IsEmpty ], 0,
function( map, elms )
return [];
end );
#############################################################################
##
#M PreImagesRange( <map> ) . . . . . . . . . . . . . . . for general mapping
##
InstallMethod( PreImagesRange,
"for general mapping",
true,
[ IsGeneralMapping ], 0,
map -> PreImagesSet( map, Range( map ) ) );
#############################################################################
##
#M PreImagesRange( <map> ) . . . . . . . . . . . . for total general mapping
##
InstallMethod( PreImagesRange,
"for total general mapping (delegate to `Source')",
true,
[ IsGeneralMapping and IsTotal ],
SUM_FLAGS, # immediately delegate, don't try anything else
Source );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> ) . . for s.p. gen. mapping & elm
##
InstallMethod( PreImagesRepresentative,
"for s.p. general mapping, and element",
FamRangeEqFamElm,
[ IsSPGeneralMapping, IsObject ], 0,
function( map, elm )
Error( "no default method for s.p. general mapping" );
end );
#############################################################################
##
#M PreImagesRepresentative( <map>, <elm> )
##
InstallMethod( PreImagesRepresentative,
"for total non-s.p. general mapping, and element",
FamRangeEqFamElm,
[ IsNonSPGeneralMapping, IsObject ], 0,
function ( map, elm )
local pres; # all preimages of <elm> under <map>
# get all preimages of <elm> under <map>
pres := PreImagesElm( map, elm );
# check that <elm> has at least one preimage under <map>
if IsEmpty( pres ) then
return fail;
fi;
# pick one preimage, and return it.
return Representative( pres );
end );
#############################################################################
##
#F GeneralMappingByElements( <S>, <R>, <elms> )
##
InstallGlobalFunction( GeneralMappingByElements, function( S, R, elms )
local map, tupfam, rel;
# Check the arguments.
if not ( IsDomain( S ) and IsDomain( R ) ) then
Error( "<S> and <R> must be domains" );
elif IsDirectProductElementCollection( elms ) then
tupfam:= ElementsFamily( FamilyObj( elms ) );
if not ( IsIdenticalObj( ElementsFamily( FamilyObj( S ) ),
ComponentsOfDirectProductElementsFamily( tupfam )[1] )
and IsIdenticalObj( ElementsFamily( FamilyObj( R ) ),
ComponentsOfDirectProductElementsFamily( tupfam )[2] ) ) then
Error( "families of arguments do not match" );
fi;
elif IsEmpty( elms ) then
tupfam:= DirectProductElementsFamily( [ ElementsFamily( FamilyObj( S ) ),
ElementsFamily( FamilyObj( R ) ) ] );
else
Error( "<elms> must be a collection of direct product elements or empty" );
fi;
# Construct the general mapping.
map:= Objectify( TypeOfDefaultGeneralMapping( S, R,
IsNonSPGeneralMapping
and IsAttributeStoringRep ),
rec() );
# Construct the underlying relation.
rel:= DomainByGenerators( tupfam, elms );
SetUnderlyingRelation( map, rel );
SetUnderlyingGeneralMapping( rel, map );
# Return the general mapping.
return map;
end );
#############################################################################
##
#M UnderlyingRelation( <map> ) . . . . . . . . . . . . for a general mapping
##
InstallMethod( UnderlyingRelation,
"for a general mapping",
true,
[ IsGeneralMapping ], 0,
function( map )
local type, rel;
type:= NewType( DirectProductFamily( [ FamilyObj( Source( map ) ),
FamilyObj( Range( map ) ) ] ),
IsDomain and IsAttributeStoringRep );
rel:= Objectify( type, rec() );
SetUnderlyingGeneralMapping( rel, map );
return rel;
end );
#############################################################################
##
#M SetUnderlyingGeneralMapping( <rel>, <map> )
##
## Make sure that <map> gets the flag `IsConstantTimeAccessGeneralMapping'
## if <rel> knows its `AsList'.
## (Note that if `AsList( <rel> )' is known at the time when <rel> is
## constructed, we cannot use the setter method of `AsList' for domains
## with known `UnderlyingGeneralMapping'.)
##
InstallMethod( SetUnderlyingGeneralMapping,
"for an underlying relation and a general mapping",
true,
[ IsDomain and IsDirectProductElementCollection and HasAsList
and IsAttributeStoringRep,
IsGeneralMapping ], 0,
function( rel, map )
SetIsConstantTimeAccessGeneralMapping( map, true );
TryNextMethod();
end );
InstallMethod( SetUnderlyingGeneralMapping,
"for an underlying relation and a general mapping",
true,
[ IsDomain and IsDirectProductElementCollection and HasGeneratorsOfDomain
and IsAttributeStoringRep,
IsGeneralMapping ], 0,
function( rel, map )
SetIsConstantTimeAccessGeneralMapping( map, true );
TryNextMethod();
end );
#############################################################################
##
#M SetAsList( <rel>, <dpelms> )
#M SetGeneratorsOfDomain( <rel>, <dpelms> )
##
## Make sure that <map> gets the flag `IsConstantTimeAccessGeneralMapping'
## if <rel> knows its `AsList' or `GeneratorsOfDomain' value,
## where <map> is the underlying general mapping of <rel>.
##
InstallMethod( SetAsList,
"for an underlying relation and a list of direct product elements",
IsIdenticalObj,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping
and IsAttributeStoringRep,
IsDirectProductElementCollection ],
function( rel, dpelms )
SetIsConstantTimeAccessGeneralMapping( UnderlyingGeneralMapping( rel ),
true );
TryNextMethod();
end );
InstallMethod( SetGeneratorsOfDomain,
"for an underlying relation and a list of direct product elements",
IsIdenticalObj,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping
and IsAttributeStoringRep,
IsDirectProductElementCollection ],
function( rel, dpelms )
SetIsConstantTimeAccessGeneralMapping( UnderlyingGeneralMapping( rel ),
true );
TryNextMethod();
end );
#############################################################################
##
## 3. generic methods for underlying relations of general mappings
##
## If the underlying relation $Rel$ of a general mapping $F$ stores $F$
## as value of `UnderlyingGeneralMapping' then $Rel$ may delegate questions
## to the mapping operations for $F$.
##
#############################################################################
##
#M \=( <rel1>, <rel2> ) . for underlying relations of two general mappings
##
InstallMethod( \=,
"for two underlying relations of general mappings",
IsIdenticalObj,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping,
IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ], 0,
function( rel1, rel2 )
local map1, map2;
map1:= UnderlyingGeneralMapping( rel1 );
map2:= UnderlyingGeneralMapping( rel2 );
# Check that the sets of first resp. second components agree.
if PreImagesRange( map1 ) <> PreImagesRange( map2 )
or ImagesSource( map1 ) <> ImagesSource( map2 ) then
return false;
fi;
# Really test the equality.
if IsFinite( PreImagesRange( map1 ) ) then
return ForAll( PreImagesRange( map1 ),
elm -> ImagesElm( map1, elm ) = ImagesElm( map2, elm ) );
elif IsFinite( PreImagesRange( map2 ) ) then
return ForAll( PreImagesRange( map2 ),
elm -> ImagesElm( map1, elm ) = ImagesElm( map2, elm ) );
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M \<( <rel1>, <rel> ) . for underlying relations of two general mappings
##
InstallMethod( \<,
"for two underlying relations of general mappings",
IsIdenticalObj,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping,
IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ], 0,
function( rel1, rel2 )
local map1, # first general mapping,
map2, # second general mapping,
elms, # elements of the source of <map1> and <map2>
i; # loop variable
map1:= UnderlyingGeneralMapping( rel1 );
map2:= UnderlyingGeneralMapping( rel2 );
# find the first element where the images differ
elms := EnumeratorSorted( Union(PreImagesRange(map1),PreImagesRange(map2)));
i := 1;
while i <= Length( elms )
and ImagesElm( map1, elms[i] ) = ImagesElm( map2, elms[i] ) do
i := i + 1;
od;
# compare the image sets
return i <= Length( elms )
and EnumeratorSorted( ImagesElm( map1, elms[i] ) )
< EnumeratorSorted( ImagesElm( map2, elms[i] ) );
#T note that we do not have a generic `\<' method for domains !
end );
#############################################################################
##
#M IsFinite( <rel> ) . . . . . for underlying relation of a general mapping
##
InstallMethod( IsFinite,
"for an underlying relation of a general mapping",
true,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ], 0,
function( rel )
local map;
map:= UnderlyingGeneralMapping( rel );
if IsFinite( Source( map ) ) and IsFinite( Range( map ) ) then
return true;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M Enumerator( <rel> ) . . . . for underlying relation of a general mapping
##
InstallMethod( Enumerator,
"for an underlying relation of a general mapping",
true,
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ], 0,
function( rel )
local map, enum, S, R, elm, imgs;
map:= UnderlyingGeneralMapping( rel );
enum:= [];
S:= Source( map );
R:= Range( map );
if IsFinite( S ) then
for elm in Enumerator( S ) do
imgs:= ImagesElm( map, elm );
if IsFinite( imgs ) then
UniteSet( enum, List( imgs, im -> DirectProductElement( [ elm, im ] ) ) );
else
TryNextMethod();
fi;
od;
return enum;
elif IsFinite( R ) then
for elm in Enumerator( R ) do
imgs:= PreImagesElm( map, elm );
if IsFinite( imgs ) then
UniteSet( enum, List( imgs, im -> DirectProductElement( [ im, elm ] ) ) );
else
TryNextMethod();
fi;
od;
return enum;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M \in( <dpelm>, <map> ) . . . . . . for elm and underl. rel. of a gen. map.
##
InstallMethod( \in,
"for an element and an underlying relation of a general mapping",
IsElmsColls,
[ IsDirectProductElement,
IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ], 0,
function( elm, rel )
return elm[2] in ImagesElm( UnderlyingGeneralMapping( rel ), elm[1] );
end );
#############################################################################
##
#M Size( <rel> ) . . . . . . . for underlying relation of a general mapping
##
InstallMethod( Size,
"for an underlying relation of a general mapping",
[ IsDomain and IsDirectProductElementCollection and HasUnderlyingGeneralMapping ],
function( rel )
local map;
map:= UnderlyingGeneralMapping( rel );
if HasIsTotal( map ) and HasIsSingleValued( map )
and IsTotal( map ) and IsSingleValued( map ) then
return Size( Source( map ) );
elif HasIsInjective( map ) and HasIsSurjective( map )
and IsInjective( map ) and IsSurjective( map ) then
return Size( Range( map ) );
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M IsGeneratorsOfMagmaWithInverses( <mappinglist> )
##
## All members of the collection have same source, all have same range.
## Check that all are invertible.
##
InstallMethod( IsGeneratorsOfMagmaWithInverses,
"for a collection of general mappings",
true,
[ IsGeneralMappingCollection ], 0,
mappinglist -> ForAll( mappinglist, map ->
(HasIsBijective(map) and IsBijective(map)) or Inverse( map ) <> fail ) );
#############################################################################
##
#M CopyMappingAttributes(<from>,<to>)
##
InstallGlobalFunction(CopyMappingAttributes,
function(f,t)
if HasIsTotal(f) and not HasIsTotal(t) then
SetIsTotal(t,IsTotal(f));
fi;
if HasIsSingleValued(f) and not HasIsSingleValued(t) then
SetIsSingleValued(t,IsSingleValued(f));
fi;
if HasIsInjective(f) and not HasIsInjective(t) then
SetIsInjective(t,IsInjective(f));
fi;
if HasIsSurjective(f) and not HasIsSurjective(t) then
SetIsSurjective(t,IsSurjective(f));
fi;
if HasSource(f) and not HasSource(t) then
SetSource(t,Source(f));
fi;
if HasImagesSource(f) and not HasImagesSource(t) then
SetImagesSource(t,ImagesSource(f));
fi;
if HasRange(f) and not HasRange(t) then
SetRange(t,Range(f));
fi;
end);
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