#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, and Heiko Theißen.
##
## 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 the definitions of properties of mappings preserving
## algebraic structure.
##
## 1. properties and attributes of gen. mappings that respect multiplication
## 2. properties and attributes of gen. mappings that respect addition
## 3. properties and attributes of gen. mappings that respect scalar mult.
## 4. properties and attributes of gen. mappings that respect multiplicative
## and additive structure
## 5. properties and attributes of gen. mappings that transform
## multiplication into addition
## 6. properties and attributes of gen. mappings that transform addition
## into multiplication
##
#############################################################################
##
## 1. properties and attributes of gen. mappings that respect multiplication
##
#############################################################################
##
#P RespectsMultiplication( <mapp> )
##
## <#GAPDoc Label="RespectsMultiplication">
## <ManSection>
## <Prop Name="RespectsMultiplication" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsMultiplication"/> returns <K>true</K> if
## <M>S</M> and <M>R</M> are magmas such that
## <M>(s_1,r_1), (s_2,r_2) \in F</M> implies
## <M>(s_1 * s_2,r_1 * r_2) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Prop="RespectsMultiplication"/> returns <K>true</K>
## if and only if the equation
## <C><A>s1</A>^<A>mapp</A> * <A>s2</A>^<A>mapp</A> =
## (<A>s1</A> * <A>s2</A>)^<A>mapp</A></C>
## holds for all <A>s1</A>, <A>s2</A> in <M>S</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsMultiplication", IsGeneralMapping );
#############################################################################
##
#P RespectsOne( <mapp> )
##
## <#GAPDoc Label="RespectsOne">
## <ManSection>
## <Prop Name="RespectsOne" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq <A>S</A> \times <A>R</A></M>,
## where <A>S</A> and <A>R</A> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsOne"/> returns <K>true</K> if
## <A>S</A> and <A>R</A> are magmas-with-one such that
## <M>( </M><C>One(<A>S</A>)</C><M>, </M><C>One(<A>R</A>)</C><M> ) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then <Ref Prop="RespectsOne"/> returns
## <K>true</K> if and only if the equation
## <C>One( <A>S</A> )^<A>mapp</A> = One( <A>R</A> )</C>
## holds.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsOne", IsGeneralMapping );
#############################################################################
##
#P RespectsInverses( <mapp> )
##
## <#GAPDoc Label="RespectsInverses">
## <ManSection>
## <Prop Name="RespectsInverses" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq <A>S</A> \times <A>R</A></M>,
## where <A>S</A> and <A>R</A> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsInverses"/> returns <K>true</K> if
## <A>S</A> and <A>R</A> are magmas-with-inverses such that,
## for <M>s \in <A>S</A></M> and <M>r \in <A>R</A></M>,
## <M>(s,r) \in F</M> implies <M>(s^{{-1}},r^{{-1}}) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then <Ref Prop="RespectsInverses"/>
## returns <K>true</K> if and only if the equation
## <C>Inverse( <A>s</A> )^<A>mapp</A> = Inverse( <A>s</A>^<A>mapp</A> )</C>
## holds for all <A>s</A> in <M>S</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsInverses", IsGeneralMapping );
#############################################################################
##
#M RespectsOne( <mapp> )
##
InstallTrueMethod( RespectsOne,
RespectsMultiplication and RespectsInverses );
#############################################################################
##
#P IsGroupGeneralMapping( <mapp> )
#P IsGroupHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsGroupGeneralMapping">
## <ManSection>
## <Filt Name="IsGroupGeneralMapping" Arg='mapp'/>
## <Filt Name="IsGroupHomomorphism" Arg='mapp'/>
##
## <Description>
## A <E>group general mapping</E> is a mapping which respects multiplication
## and inverses.
## If it is total and single valued it is called a
## <E>group homomorphism</E>.
## <P/>
## Chapter <Ref Chap="Group Homomorphisms"/> explains
## group homomorphisms in more detail.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsGroupGeneralMapping",
IsGeneralMapping and RespectsMultiplication and RespectsInverses );
DeclareSynonymAttr( "IsGroupHomomorphism",
IsGroupGeneralMapping and IsMapping );
#############################################################################
##
#A KernelOfMultiplicativeGeneralMapping( <mapp> )
##
## <#GAPDoc Label="KernelOfMultiplicativeGeneralMapping">
## <ManSection>
## <Attr Name="KernelOfMultiplicativeGeneralMapping" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping.
## Then <Ref Attr="KernelOfMultiplicativeGeneralMapping"/> returns
## the set of all elements in the source of <A>mapp</A> that have
## the identity of the range in their set of images.
## <P/>
## (This is a monoid if <A>mapp</A> respects multiplication and one,
## and if the source of <A>mapp</A> is associative.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "KernelOfMultiplicativeGeneralMapping",
IsGeneralMapping );
#############################################################################
##
#A CoKernelOfMultiplicativeGeneralMapping( <mapp> )
##
## <#GAPDoc Label="CoKernelOfMultiplicativeGeneralMapping">
## <ManSection>
## <Attr Name="CoKernelOfMultiplicativeGeneralMapping" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping.
## Then <Ref Attr="CoKernelOfMultiplicativeGeneralMapping"/> returns
## the set of all elements in the range of <A>mapp</A> that have
## the identity of the source in their set of preimages.
## <P/>
## (This is a monoid if <A>mapp</A> respects multiplication and one,
## and if the range of <A>mapp</A> is associative.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoKernelOfMultiplicativeGeneralMapping",
IsGeneralMapping );
#############################################################################
##
## 2. properties and attributes of gen. mappings that respect addition
##
#############################################################################
##
#P RespectsAddition( <mapp> )
##
## <#GAPDoc Label="RespectsAddition">
## <ManSection>
## <Prop Name="RespectsAddition" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsAddition"/> returns <K>true</K> if
## <M>S</M> and <M>R</M> are additive magmas such that
## <M>(s_1,r_1), (s_2,r_2) \in F</M> implies
## <M>(s_1 + s_2,r_1 + r_2) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then <Ref Prop="RespectsAddition"/>
## returns <K>true</K> if and only if the equation
## <C><A>s1</A>^<A>mapp</A> + <A>s2</A>^<A>mapp</A> =
## (<A>s1</A>+<A>s2</A>)^<A>mapp</A></C>
## holds for all <A>s1</A>, <A>s2</A> in <M>S</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsAddition", IsGeneralMapping );
#############################################################################
##
#P RespectsZero( <mapp> )
##
## <#GAPDoc Label="RespectsZero">
## <ManSection>
## <Prop Name="RespectsZero" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq <A>S</A> \times <A>R</A></M>,
## where <A>S</A> and <A>R</A> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsZero"/> returns <K>true</K> if
## <A>S</A> and <A>R</A> are additive-magmas-with-zero such that
## <M>( </M><C>Zero(<A>S</A>)</C><M>,
## </M><C>Zero(<A>R</A>)</C><M> ) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then <Ref Prop="RespectsZero"/> returns
## <K>true</K> if and only if the equation
## <C>Zero( <A>S</A> )^<A>mapp</A> = Zero( <A>R</A> )</C>
## holds.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsZero", IsGeneralMapping );
#############################################################################
##
#P RespectsAdditiveInverses( <mapp> )
##
## <#GAPDoc Label="RespectsAdditiveInverses">
## <ManSection>
## <Prop Name="RespectsAdditiveInverses" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsAdditiveInverses"/> returns <K>true</K> if
## <M>S</M> and <M>R</M> are additive-magmas-with-inverses such that
## <M>(s,r) \in F</M> implies <M>(-s,-r) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Prop="RespectsAdditiveInverses"/> returns <K>true</K>
## if and only if the equation
## <C>AdditiveInverse( <A>s</A> )^<A>mapp</A> =
## AdditiveInverse( <A>s</A>^<A>mapp</A> )</C>
## holds for all <A>s</A> in <M>S</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsAdditiveInverses", IsGeneralMapping );
#############################################################################
##
#M RespectsZero( <mapp> )
##
InstallTrueMethod( RespectsZero,
RespectsAddition and RespectsAdditiveInverses );
#############################################################################
##
#P IsAdditiveGroupGeneralMapping( <mapp> )
#P IsAdditiveGroupHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsAdditiveGroupGeneralMapping">
## <ManSection>
## <Filt Name="IsAdditiveGroupGeneralMapping" Arg='mapp'/>
## <Filt Name="IsAdditiveGroupHomomorphism" Arg='mapp'/>
##
## <Description>
## <Ref Filt="IsAdditiveGroupGeneralMapping"/>
## specifies whether a general mapping <A>mapp</A> respects
## addition (see <Ref Prop="RespectsAddition"/>) and respects
## additive inverses (see <Ref Prop="RespectsAdditiveInverses"/>).
## <P/>
## <Ref Filt="IsAdditiveGroupHomomorphism"/> is a synonym for the meet of
## <Ref Filt="IsAdditiveGroupGeneralMapping"/> and <Ref Filt="IsMapping"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsAdditiveGroupGeneralMapping",
IsGeneralMapping and RespectsAddition and RespectsAdditiveInverses );
DeclareSynonymAttr( "IsAdditiveGroupHomomorphism",
IsAdditiveGroupGeneralMapping and IsMapping );
#############################################################################
##
#A KernelOfAdditiveGeneralMapping( <mapp> )
##
## <#GAPDoc Label="KernelOfAdditiveGeneralMapping">
## <ManSection>
## <Attr Name="KernelOfAdditiveGeneralMapping" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping.
## Then <Ref Attr="KernelOfAdditiveGeneralMapping"/> returns
## the set of all elements in the source of <A>mapp</A> that have
## the zero of the range in their set of images.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "KernelOfAdditiveGeneralMapping", IsGeneralMapping );
#############################################################################
##
#A CoKernelOfAdditiveGeneralMapping( <mapp> )
##
## <#GAPDoc Label="CoKernelOfAdditiveGeneralMapping">
## <ManSection>
## <Attr Name="CoKernelOfAdditiveGeneralMapping" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping.
## Then <Ref Attr="CoKernelOfAdditiveGeneralMapping"/> returns
## the set of all elements in the range of <A>mapp</A> that have
## the zero of the source in their set of preimages.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CoKernelOfAdditiveGeneralMapping", IsGeneralMapping );
#############################################################################
##
## 3. properties and attributes of gen. mappings that respect scalar mult.
##
#############################################################################
##
#P RespectsScalarMultiplication( <mapp> )
##
## <#GAPDoc Label="RespectsScalarMultiplication">
## <ManSection>
## <Prop Name="RespectsScalarMultiplication" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping, with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Prop="RespectsScalarMultiplication"/> returns <K>true</K> if
## <M>S</M> and <M>R</M> are left modules with the left acting domain
## <M>D</M> of <M>S</M> contained in the left acting domain of <M>R</M>
## and such that
## <M>(s,r) \in F</M> implies <M>(c * s,c * r) \in F</M> for all
## <M>c \in D</M>, and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Prop="RespectsScalarMultiplication"/> returns
## <K>true</K> if and only if the equation
## <C><A>c</A> * <A>s</A>^<A>mapp</A> =
## (<A>c</A> * <A>s</A>)^<A>mapp</A></C>
## holds for all <A>c</A> in <M>D</M> and <A>s</A> in <M>S</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "RespectsScalarMultiplication", IsGeneralMapping );
InstallTrueMethod( RespectsAdditiveInverses, RespectsScalarMultiplication );
#############################################################################
##
#P IsLeftModuleGeneralMapping( <mapp> )
#P IsLeftModuleHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsLeftModuleGeneralMapping">
## <ManSection>
## <Filt Name="IsLeftModuleGeneralMapping" Arg='mapp'/>
## <Filt Name="IsLeftModuleHomomorphism" Arg='mapp'/>
##
## <Description>
## <Ref Filt="IsLeftModuleGeneralMapping"/>
## specifies whether a general mapping <A>mapp</A> satisfies the property
## <Ref Filt="IsAdditiveGroupGeneralMapping"/> and respects scalar
## multiplication (see <Ref Prop="RespectsScalarMultiplication"/>).
## <P/>
## <Ref Filt="IsLeftModuleHomomorphism"/> is a synonym for the meet of
## <Ref Filt="IsLeftModuleGeneralMapping"/> and <Ref Filt="IsMapping"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsLeftModuleGeneralMapping",
IsAdditiveGroupGeneralMapping and RespectsScalarMultiplication );
DeclareSynonymAttr( "IsLeftModuleHomomorphism",
IsLeftModuleGeneralMapping and IsMapping );
#############################################################################
##
#O IsLinearMapping( <F>, <mapp> )
##
## <#GAPDoc Label="IsLinearMapping">
## <ManSection>
## <Oper Name="IsLinearMapping" Arg='F, mapp'/>
##
## <Description>
## For a field <A>F</A> and a general mapping <A>mapp</A>,
## <Ref Oper="IsLinearMapping"/> returns <K>true</K> if <A>mapp</A> is an
## <A>F</A>-linear mapping, and <K>false</K> otherwise.
## <P/>
## A mapping <M>f</M> is a linear mapping (or vector space homomorphism)
## if the source and range are vector spaces over the same division ring
## <M>D</M>, and if
## <M>f( a + b ) = f(a) + f(b)</M> and <M>f( s * a ) = s * f(a)</M> hold
## for all elements <M>a</M>, <M>b</M> in the source of <M>f</M>
## and <M>s \in D</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "IsLinearMapping", [ IsDomain, IsGeneralMapping ] );
#############################################################################
##
## 4. properties and attributes of gen. mappings that respect multiplicative
## and additive structure
##
#############################################################################
##
#P IsRingGeneralMapping( <mapp> )
#P IsRingHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsRingGeneralMapping">
## <ManSection>
## <Filt Name="IsRingGeneralMapping" Arg='mapp'/>
## <Filt Name="IsRingHomomorphism" Arg='mapp'/>
##
## <Description>
## <Ref Filt="IsRingGeneralMapping"/> specifies whether a general mapping
## <A>mapp</A> satisfies the property
## <Ref Filt="IsAdditiveGroupGeneralMapping"/> and respects multiplication
## (see <Ref Prop="RespectsMultiplication"/>).
## <P/>
## <Ref Filt="IsRingHomomorphism"/> is a synonym for the meet of
## <Ref Filt="IsRingGeneralMapping"/> and <Ref Filt="IsMapping"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsRingGeneralMapping",
IsGeneralMapping and RespectsMultiplication
and IsAdditiveGroupGeneralMapping );
DeclareSynonymAttr( "IsRingHomomorphism",
IsRingGeneralMapping and IsMapping );
#############################################################################
##
#P IsRingWithOneGeneralMapping( <mapp> )
#P IsRingWithOneHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsRingWithOneGeneralMapping">
## <ManSection>
## <Filt Name="IsRingWithOneGeneralMapping" Arg='mapp'/>
## <Filt Name="IsRingWithOneHomomorphism" Arg='mapp'/>
##
## <Description>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsRingWithOneGeneralMapping",
IsRingGeneralMapping and RespectsOne );
DeclareSynonymAttr( "IsRingWithOneHomomorphism",
IsRingWithOneGeneralMapping and IsMapping );
#############################################################################
##
#P IsAlgebraGeneralMapping( <mapp> )
#P IsAlgebraHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsAlgebraGeneralMapping">
## <ManSection>
## <Filt Name="IsAlgebraGeneralMapping" Arg='mapp'/>
## <Filt Name="IsAlgebraHomomorphism" Arg='mapp'/>
##
## <Description>
## <Ref Filt="IsAlgebraGeneralMapping"/> specifies whether a general
## mapping <A>mapp</A> satisfies both properties
## <Ref Filt="IsRingGeneralMapping"/> and
## (see <Ref Filt="IsLeftModuleGeneralMapping"/>).
## <P/>
## <Ref Filt="IsAlgebraHomomorphism"/> is a synonym for the meet of
## <Ref Filt="IsAlgebraGeneralMapping"/> and <Ref Filt="IsMapping"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsAlgebraGeneralMapping",
IsRingGeneralMapping and IsLeftModuleGeneralMapping );
DeclareSynonymAttr( "IsAlgebraHomomorphism",
IsAlgebraGeneralMapping and IsMapping );
#############################################################################
##
#P IsAlgebraWithOneGeneralMapping( <mapp> )
#P IsAlgebraWithOneHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsAlgebraWithOneGeneralMapping">
## <ManSection>
## <Filt Name="IsAlgebraWithOneGeneralMapping" Arg='mapp'/>
## <Filt Name="IsAlgebraWithOneHomomorphism" Arg='mapp'/>
##
## <Description>
## <Ref Filt="IsAlgebraWithOneGeneralMapping"/>
## specifies whether a general mapping <A>mapp</A> satisfies both
## properties <Ref Filt="IsAlgebraGeneralMapping"/> and
## <Ref Prop="RespectsOne"/>.
## <P/>
## <Ref Filt="IsAlgebraWithOneHomomorphism"/> is a synonym for the meet of
## <Ref Filt="IsAlgebraWithOneGeneralMapping"/> and <Ref Filt="IsMapping"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsAlgebraWithOneGeneralMapping",
IsAlgebraGeneralMapping and RespectsOne );
DeclareSynonymAttr( "IsAlgebraWithOneHomomorphism",
IsAlgebraWithOneGeneralMapping and IsMapping );
#############################################################################
##
#P IsFieldHomomorphism( <mapp> )
##
## <#GAPDoc Label="IsFieldHomomorphism">
## <ManSection>
## <Prop Name="IsFieldHomomorphism" Arg='mapp'/>
##
## <Description>
## A general mapping is a field homomorphism if and only if it is
## a ring homomorphism with source a field.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFieldHomomorphism", IsGeneralMapping );
InstallTrueMethod( IsAlgebraHomomorphism, IsFieldHomomorphism );
#############################################################################
##
#F InstallEqMethodForMappingsFromGenerators( <IsStruct>,
#F <GeneratorsOfStruct>, <respects>, <infostring> )
##
## <ManSection>
## <Func Name="InstallEqMethodForMappingsFromGenerators"
## Arg='IsStruct, GeneratorsOfStruct, respects, infostring'/>
##
## <Description>
## </Description>
## </ManSection>
##
BindGlobal( "InstallEqMethodForMappingsFromGenerators", function( IsStruct,
GeneratorsOfStruct, respects, infostring )
InstallMethod( \=,
Concatenation( "method for two s.v. gen. mappings", infostring ),
IsIdenticalObj,
[ IsGeneralMapping and IsSingleValued and respects,
IsGeneralMapping and IsSingleValued and respects ],
0,
function( map1, map2 )
local preim, gen;
if not IsStruct( Source( map1 ) ) then
TryNextMethod();
elif HasIsInjective( map1 ) and HasIsInjective( map2 )
and IsInjective( map1 ) <> IsInjective( map2 ) then
return false;
elif HasIsSurjective( map1 ) and HasIsSurjective( map2 )
and IsSurjective( map1 ) <> IsSurjective( map2 ) then
return false;
elif HasIsTotal( map1 ) and HasIsTotal( map2 )
and IsTotal( map1 ) <> IsTotal( map2 ) then
return false;
elif Source( map1 ) <> Source( map2 )
or Range ( map1 ) <> Range ( map2 ) then
return false;
fi;
preim:= PreImagesRange( map1 );
if not IsStruct( preim ) then
TryNextMethod();
fi;
for gen in GeneratorsOfStruct( preim ) do
if ImagesRepresentative( map1, gen )
<> ImagesRepresentative( map2, gen ) then
return false;
fi;
od;
return true;
end );
InstallMethod(IsOne,
Concatenation( "method for s.v. gen. mapping", infostring ),
true,
[ IsGeneralMapping and IsSingleValued and respects ],
0,
function( map )
local gen;
if not IsStruct( Source( map ) ) then
TryNextMethod();
elif HasIsInjective( map ) and not IsInjective( map ) then
return false;
elif HasIsSurjective( map ) and not IsSurjective( map ) then
return false;
elif HasIsTotal( map ) and not IsTotal( map ) then
return false;
elif Source( map ) <> Range( map ) then
return false;
fi;
for gen in GeneratorsOfStruct( Source(map) ) do
if gen<>ImagesRepresentative( map, gen ) then
return false;
fi;
od;
return true;
end );
end );
#############################################################################
##
## 5. properties and attributes of gen. mappings that transform
## multiplication into addition
##
#############################################################################
##
#P TransformsMultiplicationIntoAddition( <mapp> )
##
## <ManSection>
## <Prop Name="TransformsMultiplicationIntoAddition" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="TransformsMultiplicationIntoAddition"/> returns
## <K>true</K> if <M>S</M> is a magma and <M>R</M> an additive magma
## such that <M>(s_1,r_1), (s_2,r_2) \in F</M> implies
## <M>(s_1 * s_2,r_1 + r_2) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TransformsMultiplicationIntoAddition"/>
## returns <K>true</K> if and only if the equation
## <C><A>s1</A>^<A>mapp</A> + <A>s2</A>^<A>mapp</A> =
## (<A>s1</A> * <A>s2</A>)^<A>mapp</A></C> holds for all
## <A>s1</A>, <A>s2</A> in <M>S</M>.
## </Description>
## </ManSection>
##
DeclareProperty( "TransformsMultiplicationIntoAddition", IsGeneralMapping );
#############################################################################
##
#P TranformsOneIntoZero( <mapp> )
##
## <ManSection>
## <Prop Name="TranformsOneIntoZero" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="TranformsOneIntoZero"/> returns <K>true</K> if
## <M>S</M> is a magma-with-one and <M>R</M> an additive-magma-with-zero
## such that
## <M>( </M><C>One(</C><M>S</M><C>), Zero(</C><M>R</M><C>) )</C><M> \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TranformsOneIntoZero"/> returns <K>true</K>
## if and only if the equation
## <C>One( S )^<A>mapp</A> = Zero( R )</C>
## holds.
## </Description>
## </ManSection>
##
DeclareProperty( "TranformsOneIntoZero", IsGeneralMapping );
#############################################################################
##
#P TransformsInversesIntoAdditiveInverses( <mapp> )
##
## <ManSection>
## <Prop Name="TransformsInversesIntoAdditiveInverses" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="RespectsInverses"/> returns <K>true</K> if
## <M>S</M> and <M>R</M> are magmas-with-inverses such that
## <M>S</M> is a magma-with-inverses
## and <M>R</M> an additive-magma-with-inverses
## such that <M>(s,r) \in F</M> implies <M>(s^{-1},-r) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TransformsInversesIntoAdditiveInverses"/>
## returns <K>true</K> if and only if the equation
## <C>Inverse( <A>s</A> )^<A>mapp</A> =
## AdditiveInverse( <A>s</A>^<A>mapp</A> )</C>
## holds for all <A>s</A> in <M>S</M>.
## </Description>
## </ManSection>
##
DeclareProperty( "TransformsInversesIntoAdditiveInverses", IsGeneralMapping );
#############################################################################
##
#M RespectsOne( <mapp> )
##
InstallTrueMethod( TranformsOneIntoZero,
TransformsMultiplicationIntoAddition and
TransformsInversesIntoAdditiveInverses );
#############################################################################
##
#P IsGroupToAdditiveGroupGeneralMapping( <mapp> )
#P IsGroupToAdditiveGroupHomomorphism( <mapp> )
##
## <ManSection>
## <Prop Name="IsGroupToAdditiveGroupGeneralMapping" Arg='mapp'/>
## <Prop Name="IsGroupToAdditiveGroupHomomorphism" Arg='mapp'/>
##
## <Description>
## A <C>GroupToAdditiveGroupGeneralMapping</C> is a mapping which transforms
## multiplication into addition and transforms
## inverses into additive inverses. If it is total and single valued it is
## called a group-to-additive-group
## homomorphism.
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsGroupToAdditiveGroupGeneralMapping",
IsGeneralMapping and TransformsMultiplicationIntoAddition and
TransformsInversesIntoAdditiveInverses );
DeclareSynonymAttr( "IsGroupToAdditiveGroupHomomorphism",
IsGroupToAdditiveGroupGeneralMapping and IsMapping );
#############################################################################
##
## 6. properties and attributes of gen. mappings that transform addition
## into multiplication
##
#############################################################################
##
#P TransformsAdditionIntoMultiplication( <mapp> )
##
## <ManSection>
## <Prop Name="TransformsAdditionIntoMultiplication" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="TransformsAdditionIntoMultiplication"/> returns
## <K>true</K> if
## <M>S</M> is an additive magma and <M>R</M> a magma such that
## <M>(s_1,r_1), (s_2,r_2) \in F</M> implies <M>(s_1 + s_2,r_1 * r_2) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TransformsAdditionIntoMultiplication"/>
## returns <K>true</K> if and only if the equation
## <C><A>s1</A>^<A>mapp</A> * <A>s2</A>^<A>mapp</A> =
## (<A>s1</A>+<A>s2</A>)^<A>mapp</A></C>
## holds for all <A>s1</A>, <A>s2</A> in <M>S</M>.
## </Description>
## </ManSection>
##
DeclareProperty( "TransformsAdditionIntoMultiplication", IsGeneralMapping );
#############################################################################
##
#P TransformsZeroIntoOne( <mapp> )
##
## <ManSection>
## <Prop Name="TransformsZeroIntoOne" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="TransformsZeroIntoOne"/> returns <K>true</K> if
## <M>S</M> is an additive-magma-with-zero and <M>R</M> a magma-with-one
## such that
## <M>( </M><C>Zero(</C><M>S</M><C>), One(</C><M>R</M><C>) )</C><M> \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TransformsZeroIntoOne"/> returns <K>true</K>
## if and only if the equation
## <C>Zero( S )^<A>mapp</A> = One( R )</C>
## holds.
## </Description>
## </ManSection>
##
DeclareProperty( "TransformsZeroIntoOne", IsGeneralMapping );
#############################################################################
##
#P TransformsAdditiveInversesIntoInverses( <mapp> )
##
## <ManSection>
## <Prop Name="TransformsAdditiveInversesIntoInverses" Arg='mapp'/>
##
## <Description>
## Let <A>mapp</A> be a general mapping with underlying relation
## <M>F \subseteq S \times R</M>,
## where <M>S</M> and <M>R</M> are the source and the range of <A>mapp</A>,
## respectively.
## Then <Ref Func="TransformsAdditiveInversesIntoInverses"/> returns
## <K>true</K> if <M>S</M> is an additive-magma-with-inverses and
## <M>R</M> a magma-with-inverses such that
## <M>(s,r) \in F</M> implies <M>(-s,r^{-1}) \in F</M>,
## and <K>false</K> otherwise.
## <P/>
## If <A>mapp</A> is single-valued then
## <Ref Func="TransformsAdditiveInversesIntoInverses"/>
## returns <K>true</K> if and only if the equation
## <C>AdditiveInverse( <A>s</A> )^<A>mapp</A> =
## Inverse( <A>s</A>^<A>mapp</A> )</C> holds for all <A>s</A> in <M>S</M>.
## </Description>
## </ManSection>
##
DeclareProperty( "TransformsAdditiveInversesIntoInverses", IsGeneralMapping );
#############################################################################
##
#M TransformsAdditiveInversesIntoInverses( <mapp> )
##
InstallTrueMethod( TransformsAdditiveInversesIntoInverses,
TransformsAdditionIntoMultiplication and
TransformsAdditiveInversesIntoInverses );
#############################################################################
##
#P IsAdditiveGroupToGroupGeneralMapping( <mapp> )
#P IsAdditiveGroupToGroupHomomorphism( <mapp> )
##
## <ManSection>
## <Prop Name="IsAdditiveGroupToGroupGeneralMapping" Arg='mapp'/>
## <Prop Name="IsAdditiveGroupToGroupHomomorphism" Arg='mapp'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsAdditiveGroupToGroupGeneralMapping",
IsGeneralMapping and TransformsAdditionIntoMultiplication and
TransformsAdditiveInversesIntoInverses );
DeclareSynonymAttr( "IsAdditiveGroupToGroupHomomorphism",
IsAdditiveGroupToGroupGeneralMapping and IsMapping );
