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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer.
##
## 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 declares the operations for additive magmas,
## Note that the meaning of generators for the three categories
## additive magma, additive-magma-with-zero,
## and additive-magma-with-inverses is different.
##
#############################################################################
##
#C IsNearAdditiveMagma( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveMagma">
## <ManSection>
## <Filt Name="IsNearAdditiveMagma" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive magma</E> in &GAP; is a domain <M>A</M>
## with an associative but not necessarily commutative addition
## <C>+</C><M>: A \times A \rightarrow A</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveMagma",
IsDomain and IsNearAdditiveElementCollection );
#############################################################################
##
#C IsNearAdditiveMagmaWithZero( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveMagmaWithZero">
## <ManSection>
## <Filt Name="IsNearAdditiveMagmaWithZero" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive magma-with-zero</E> in &GAP; is a near-additive magma
## <M>A</M> with an operation <C>0*</C> (or <Ref Attr="Zero"/>)
## that yields the zero element of <M>A</M>.
## <P/>
## So a near-additive magma-with-zero <A>A</A> does always contain a unique
## additively neutral element <M>z</M>,
## i.e., <M>z + a = a = a + z</M> holds for all <M>a \in A</M>
## (see <Ref Attr="AdditiveNeutralElement"/>).
## This zero element <M>z</M> can be computed with the operation
## <Ref Attr="Zero"/>, by applying this function to <M>A</M> or to any
## element <M>a</M> in <M>A</M>.
## The zero element can be computed also as <C>0 * </C><M>a</M>,
## for any <M>a</M> in <M>A</M>.
## <P/>
## <E>Note</E> that it may happen that
## a near-additive magma containing a zero does <E>not</E> lie in the
## category <Ref Filt="IsNearAdditiveMagmaWithZero"/>
## (see <Ref Sect="Domain Categories"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveMagmaWithZero",
IsNearAdditiveMagma and IsNearAdditiveElementWithZeroCollection );
#############################################################################
##
#C IsNearAdditiveGroup( <obj> )
#C IsNearAdditiveMagmaWithInverses( <obj> )
##
## <#GAPDoc Label="IsNearAdditiveGroup">
## <ManSection>
## <Filt Name="IsNearAdditiveGroup" Arg='obj' Type='Category'/>
## <Filt Name="IsNearAdditiveMagmaWithInverses" Arg='obj' Type='Category'/>
##
## <Description>
## A <E>near-additive group</E> in &GAP; is a near-additive magma-with-zero
## <M>A</M> with an operation <C>-1*</C><M>: A \rightarrow A</M> that maps
## each element <M>a</M> of <M>A</M> to its additive inverse
## <C>-1*</C><M>a</M> (or <C>AdditiveInverse( </C><A>a</A><C> )</C>,
## see <Ref Attr="AdditiveInverse"/>).
## <P/>
## The addition <C>+</C> of <M>A</M> is assumed to be associative,
## so a near-additive group is not more than a
## <E>near-additive magma-with-inverses</E>.
## <Ref Filt="IsNearAdditiveMagmaWithInverses"/> is just a synonym for
## <Ref Filt="IsNearAdditiveGroup"/>,
## and can be used alternatively in all function names involving the string
## <C>"NearAdditiveGroup"</C>.
## <P/>
## Note that not every trivial near-additive magma is a near-additive
## magma-with-zero,
## but every trivial near-additive magma-with-zero is a near-additive group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsNearAdditiveGroup",
IsNearAdditiveMagmaWithZero
and IsNearAdditiveElementWithInverseCollection );
DeclareSynonym( "IsNearAdditiveMagmaWithInverses", IsNearAdditiveGroup );
#############################################################################
##
#P IsAdditivelyCommutative( <A> )
##
## <#GAPDoc Label="IsAdditivelyCommutative">
## <ManSection>
## <Prop Name="IsAdditivelyCommutative" Arg='A'/>
##
## <Description>
## A near-additive magma <A>A</A> in &GAP; is <E>additively commutative</E>
## if for all elements <M>a, b \in <A>A</A></M> the equality
## <M>a + b = b + a</M> holds.
## <P/>
## Note that the commutativity of the <E>multiplication</E> <C>*</C> in a
## multiplicative structure can be tested with <Ref Prop="IsCommutative"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsAdditivelyCommutative", IsNearAdditiveMagma );
InstallTrueMethod( IsAdditivelyCommutative,
IsAdditivelyCommutativeElementCollection and IsMagma );
InstallSubsetMaintenance( IsAdditivelyCommutative,
IsNearAdditiveMagma and IsAdditivelyCommutative, IsNearAdditiveMagma );
InstallFactorMaintenance( IsAdditivelyCommutative,
IsNearAdditiveMagma and IsAdditivelyCommutative,
IsObject, IsNearAdditiveMagma );
InstallTrueMethod( IsAdditivelyCommutative,
IsNearAdditiveMagma and IsTrivial );
InstallTrueMethod( IsAdditiveElementCollection,
IsNearAdditiveElementCollection and IsAdditivelyCommutative );
InstallTrueMethod( IsAdditiveElementWithZeroCollection,
IsNearAdditiveElementWithZeroCollection and IsAdditivelyCommutative );
InstallTrueMethod( IsAdditiveElementWithInverseCollection,
IsNearAdditiveElementWithInverseCollection
and IsAdditivelyCommutative );
#############################################################################
##
#C IsAdditiveMagma( <obj> )
##
## <#GAPDoc Label="IsAdditiveMagma">
## <ManSection>
## <Filt Name="IsAdditiveMagma" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive magma</E> in &GAP; is a domain <M>A</M> with an
## associative and commutative addition
## <C>+</C><M>: A \times A \rightarrow A</M>,
## see <Ref Filt="IsNearAdditiveMagma"/> and
## <Ref Prop="IsAdditivelyCommutative"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsAdditiveMagma",
IsNearAdditiveMagma and IsAdditivelyCommutative );
#############################################################################
##
#C IsAdditiveMagmaWithZero( <obj> )
##
## <#GAPDoc Label="IsAdditiveMagmaWithZero">
## <ManSection>
## <Filt Name="IsAdditiveMagmaWithZero" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive magma-with-zero</E> in &GAP; is an additive magma <M>A</M>
## (see <Ref Filt="IsAdditiveMagma"/> with an operation <C>0*</C>
## (or <Ref Attr="Zero"/>) that yields the zero of <M>A</M>.
## <P/>
## So an additive magma-with-zero <M>A</M> does always contain a unique
## additively neutral element <M>z</M>, i.e.,
## <M>z + a = a = a + z</M> holds for all <M>a \in A</M>
## (see <Ref Attr="AdditiveNeutralElement"/>).
## This element <M>z</M> can be computed with the operation
## <Ref Attr="Zero"/> as <C>Zero( </C><M>A</M><C> )</C>,
## and <M>z</M> is also equal to <C>Zero( </C><M>a</M><C> )</C> and to
## <C>0*</C><M>a</M> for each element <M>a</M> in <M>A</M>.
## <P/>
## <E>Note</E> that it may happen that
## an additive magma containing a zero does <E>not</E> lie in the category
## <Ref Filt="IsAdditiveMagmaWithZero"/>
## (see <Ref Sect="Domain Categories"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsAdditiveMagmaWithZero",
IsNearAdditiveMagmaWithZero and IsAdditiveMagma );
#############################################################################
##
#C IsAdditiveGroup( <obj> )
#C IsAdditiveMagmaWithInverses( <obj> )
##
## <#GAPDoc Label="IsAdditiveGroup">
## <ManSection>
## <Filt Name="IsAdditiveGroup" Arg='obj' Type='Category'/>
## <Filt Name="IsAdditiveMagmaWithInverses" Arg='obj' Type='Category'/>
##
## <Description>
## An <E>additive group</E> in &GAP; is an additive magma-with-zero <M>A</M>
## with an operation <C>-1*</C><M>: A \rightarrow A</M> that maps
## each element <M>a</M> of <M>A</M> to its additive inverse
## <C>-1*</C><M>a</M> (or <C>AdditiveInverse( </C><M>a</M><C> )</C>,
## see <Ref Attr="AdditiveInverse"/>).
## <P/>
## The addition <C>+</C> of <M>A</M> is assumed to be commutative and
## associative, so an additive group is not more than an
## <E>additive magma-with-inverses</E>.
## <Ref Filt="IsAdditiveMagmaWithInverses"/> is just a synonym for
## <Ref Filt="IsAdditiveGroup"/>,
## and can be used alternatively in all function names involving the string
## <C>"AdditiveGroup"</C>.
## <P/>
## Note that not every trivial additive magma is an additive
## magma-with-zero,
## but every trivial additive magma-with-zero is an additive group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsAdditiveGroup",
IsNearAdditiveGroup and IsAdditiveMagma );
DeclareSynonym( "IsAdditiveMagmaWithInverses", IsAdditiveGroup );
#############################################################################
##
#A Zero( <D> )
##
## (see the description in `arith.gd')
##
DeclareAttribute( "Zero", IsDomain and IsAdditiveElementWithZeroCollection );
#############################################################################
##
#F NearAdditiveMagma( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveMagma">
## <ManSection>
## <Func Name="NearAdditiveMagma" Arg='[Fam, ]gens'/>
##
## <Description>
## returns the (near-)additive magma <M>A</M> that is generated by the
## elements in the list <A>gens</A>, that is,
## the closure of <A>gens</A> under addition <C>+</C>.
## The family <A>Fam</A> of <M>A</M> can be entered as first argument;
## this is obligatory if <A>gens</A> is empty
## (and hence also <M>A</M> is empty).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "NearAdditiveMagma" );
DeclareSynonym( "AdditiveMagma", NearAdditiveMagma );
#############################################################################
##
#F NearAdditiveMagmaWithZero( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveMagmaWithZero">
## <ManSection>
## <Func Name="NearAdditiveMagmaWithZero" Arg='[Fam, ]gens'/>
##
## <Description>
## returns the (near-)additive magma-with-zero <M>A</M> that is generated by
## the elements in the list <A>gens</A>, that is,
## the closure of <A>gens</A> under addition <C>+</C> and
## <Ref Attr="Zero"/>.
## The family <A>Fam</A> of <M>A</M> can be entered as first argument;
## this is obligatory if <A>gens</A> is empty
## (and hence <M>A</M> is trivial).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "NearAdditiveMagmaWithZero" );
DeclareSynonym( "AdditiveMagmaWithZero", NearAdditiveMagmaWithZero );
#############################################################################
##
#F NearAdditiveGroup( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveGroup">
## <ManSection>
## <Func Name="NearAdditiveGroup" Arg='[Fam, ]gens'/>
##
## <Description>
## returns the (near-)additive group <M>A</M> that is generated by
## the elements in the list <A>gens</A>, that is,
## the closure of <A>gens</A> under addition <C>+</C>, <Ref Attr="Zero"/>,
## and <Ref Attr="AdditiveInverse"/>.
## The family <A>Fam</A> of <M>A</M> can be entered as first argument;
## this is obligatory if <A>gens</A> is empty
## (and hence <M>A</M> is trivial).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "NearAdditiveGroup" );
DeclareSynonym( "AdditiveGroup", NearAdditiveGroup );
DeclareSynonym( "NearAdditiveMagmaWithInverses", NearAdditiveGroup );
DeclareSynonym( "AdditiveMagmaWithInverses", NearAdditiveGroup );
#############################################################################
##
#O NearAdditiveMagmaByGenerators( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveMagmaByGenerators">
## <ManSection>
## <Oper Name="NearAdditiveMagmaByGenerators" Arg='[Fam, ]gens'/>
##
## <Description>
## An underlying operation for <Ref Func="NearAdditiveMagma"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "NearAdditiveMagmaByGenerators", [ IsCollection ] );
DeclareSynonym( "AdditiveMagmaByGenerators", NearAdditiveMagmaByGenerators );
#############################################################################
##
#O NearAdditiveMagmaWithZeroByGenerators( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveMagmaWithZeroByGenerators">
## <ManSection>
## <Oper Name="NearAdditiveMagmaWithZeroByGenerators" Arg='[Fam, ]gens'/>
##
## <Description>
## An underlying operation for <Ref Func="NearAdditiveMagmaWithZero"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "NearAdditiveMagmaWithZeroByGenerators",
[ IsCollection ] );
DeclareSynonym( "AdditiveMagmaWithZeroByGenerators",
NearAdditiveMagmaWithZeroByGenerators );
#############################################################################
##
#O NearAdditiveGroupByGenerators( [<Fam>, ]<gens> )
##
## <#GAPDoc Label="NearAdditiveGroupByGenerators">
## <ManSection>
## <Oper Name="NearAdditiveGroupByGenerators" Arg='[Fam, ]gens'/>
##
## <Description>
## An underlying operation for <Ref Func="NearAdditiveGroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "NearAdditiveGroupByGenerators", [ IsCollection ] );
DeclareSynonym( "AdditiveGroupByGenerators",
NearAdditiveGroupByGenerators );
DeclareSynonym( "NearAdditiveMagmaWithInversesByGenerators",
NearAdditiveGroupByGenerators );
DeclareSynonym( "AdditiveMagmaWithInversesByGenerators",
NearAdditiveGroupByGenerators );
#############################################################################
##
#F SubnearAdditiveMagma( <D>, <gens> )
#F SubnearAdditiveMagma( <D>, <gens> )
#F SubadditiveMagmaNC( <D>, <gens> )
#F SubadditiveMagmaNC( <D>, <gens> )
##
## <#GAPDoc Label="SubnearAdditiveMagma">
## <ManSection>
## <Func Name="SubnearAdditiveMagma" Arg='D, gens'/>
## <Func Name="SubadditiveMagma" Arg='D, gens'/>
## <Func Name="SubnearAdditiveMagmaNC" Arg='D, gens'/>
## <Func Name="SubadditiveMagmaNC" Arg='D, gens'/>
##
## <Description>
## <Ref Func="SubnearAdditiveMagma"/> returns the near-additive magma
## generated by the elements in the list <A>gens</A>,
## with parent the domain <A>D</A>.
## <Ref Func="SubnearAdditiveMagmaNC"/> does the same, except that it
## does not check whether the elements of <A>gens</A> lie in <A>D</A>.
## <P/>
## <Ref Func="SubadditiveMagma"/> and <Ref Func="SubadditiveMagmaNC"/>
## are just synonyms of these functions.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SubnearAdditiveMagma" );
DeclareGlobalFunction( "SubnearAdditiveMagmaNC" );
DeclareSynonym( "SubadditiveMagma", SubnearAdditiveMagma );
DeclareSynonym( "SubadditiveMagmaNC", SubnearAdditiveMagmaNC );
#############################################################################
##
#F SubnearAdditiveMagmaWithZero( <D>, <gens> )
#F SubnearAdditiveMagmaWithZeroNC( <D>, <gens> )
##
## <#GAPDoc Label="SubnearAdditiveMagmaWithZero">
## <ManSection>
## <Func Name="SubnearAdditiveMagmaWithZero" Arg='D, gens'/>
## <Func Name="SubadditiveMagmaWithZero" Arg='D, gens'/>
## <Func Name="SubnearAdditiveMagmaWithZeroNC" Arg='D, gens'/>
## <Func Name="SubadditiveMagmaWithZeroNC" Arg='D, gens'/>
##
## <Description>
## <Ref Func="SubnearAdditiveMagmaWithZero"/> returns the near-additive
## magma-with-zero generated by the elements in the list <A>gens</A>,
## with parent the domain <A>D</A>.
## <Ref Func="SubnearAdditiveMagmaWithZeroNC"/> does the same, except that
## it does not check whether the elements of <A>gens</A> lie in <A>D</A>.
## <P/>
## <Ref Func="SubadditiveMagmaWithZero"/> and
## <Ref Func="SubadditiveMagmaWithZeroNC"/>
## are just synonyms of these functions.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SubnearAdditiveMagmaWithZero" );
DeclareGlobalFunction( "SubnearAdditiveMagmaWithZeroNC" );
DeclareSynonym( "SubadditiveMagmaWithZero", SubnearAdditiveMagmaWithZero );
DeclareSynonym( "SubadditiveMagmaWithZeroNC",
SubnearAdditiveMagmaWithZeroNC );
#############################################################################
##
#F SubnearAdditiveGroup( <D>, <gens> )
#F SubnearAdditiveGroupNC( <D>, <gens> )
##
## <#GAPDoc Label="SubnearAdditiveGroup">
## <ManSection>
## <Func Name="SubnearAdditiveGroup" Arg='D, gens'/>
## <Func Name="SubadditiveGroup" Arg='D, gens'/>
## <Func Name="SubnearAdditiveGroupNC" Arg='D, gens'/>
## <Func Name="SubadditiveGroupNC" Arg='D, gens'/>
##
## <Description>
## <Ref Func="SubnearAdditiveGroup"/> returns the near-additive group
## generated by the elements in the list <A>gens</A>,
## with parent the domain <A>D</A>.
## <Ref Func="SubadditiveGroupNC"/> does the same, except that it does not
## check whether the elements of <A>gens</A> lie in <A>D</A>.
## <P/>
## <Ref Func="SubadditiveGroup"/> and <Ref Func="SubadditiveGroupNC"/>
## are just synonyms of these functions.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SubnearAdditiveGroup" );
DeclareGlobalFunction( "SubnearAdditiveGroupNC" );
DeclareSynonym( "SubadditiveGroup", SubnearAdditiveGroup );
DeclareSynonym( "SubnearAdditiveMagmaWithInverses", SubnearAdditiveGroup );
DeclareSynonym( "SubadditiveMagmaWithInverses", SubnearAdditiveGroup );
DeclareSynonym( "SubadditiveGroupNC", SubnearAdditiveGroupNC );
DeclareSynonym( "SubnearAdditiveMagmaWithInversesNC",
SubnearAdditiveGroupNC );
DeclareSynonym( "SubadditiveMagmaWithInversesNC", SubnearAdditiveGroupNC );
#############################################################################
##
#A GeneratorsOfNearAdditiveMagma( <A> )
#A GeneratorsOfAdditiveMagma( <A> )
##
## <#GAPDoc Label="GeneratorsOfNearAdditiveMagma">
## <ManSection>
## <Attr Name="GeneratorsOfNearAdditiveMagma" Arg='A'/>
## <Attr Name="GeneratorsOfAdditiveMagma" Arg='A'/>
##
## <Description>
## is a list of elements of the near-additive magma <A>A</A>
## that generates <A>A</A> as a near-additive magma,
## that is, the closure of this list under addition is <A>A</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfNearAdditiveMagma", IsNearAdditiveMagma );
DeclareSynonymAttr( "GeneratorsOfAdditiveMagma",
GeneratorsOfNearAdditiveMagma );
#############################################################################
##
#A GeneratorsOfNearAdditiveMagmaWithZero( <A> )
#A GeneratorsOfAdditiveMagmaWithZero( <A> )
##
## <#GAPDoc Label="GeneratorsOfNearAdditiveMagmaWithZero">
## <ManSection>
## <Attr Name="GeneratorsOfNearAdditiveMagmaWithZero" Arg='A'/>
## <Attr Name="GeneratorsOfAdditiveMagmaWithZero" Arg='A'/>
##
## <Description>
## is a list of elements of the near-additive magma-with-zero
## <A>A</A> that generates <A>A</A> as a near-additive magma-with-zero,
## that is,
## the closure of this list under addition and <Ref Attr="Zero"/>
## is <A>A</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfNearAdditiveMagmaWithZero",
IsNearAdditiveMagmaWithZero );
DeclareSynonymAttr( "GeneratorsOfAdditiveMagmaWithZero",
GeneratorsOfNearAdditiveMagmaWithZero );
#############################################################################
##
#A GeneratorsOfNearAdditiveGroup( <A> )
#A GeneratorsOfAdditiveGroup( <A> )
##
## <#GAPDoc Label="GeneratorsOfNearAdditiveGroup">
## <ManSection>
## <Attr Name="GeneratorsOfNearAdditiveGroup" Arg='A'/>
## <Attr Name="GeneratorsOfAdditiveGroup" Arg='A'/>
##
## <Description>
## is a list of elements of the near-additive group <A>A</A>
## that generates <A>A</A> as a near-additive group,
## that is, the closure of this list under addition,
## taking the zero element, and taking additive inverses
## (see <Ref Attr="AdditiveInverse"/>) is <A>A</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "GeneratorsOfNearAdditiveGroup", IsNearAdditiveGroup );
DeclareSynonymAttr( "GeneratorsOfAdditiveMagmaWithInverses",
GeneratorsOfNearAdditiveGroup );
DeclareSynonymAttr( "GeneratorsOfNearAdditiveMagmaWithInverses",
GeneratorsOfNearAdditiveGroup );
DeclareSynonymAttr( "GeneratorsOfAdditiveGroup",
GeneratorsOfNearAdditiveGroup );
#############################################################################
##
#A TrivialSubnearAdditiveMagmaWithZero( <A> )
##
## <#GAPDoc Label="TrivialSubnearAdditiveMagmaWithZero">
## <ManSection>
## <Attr Name="TrivialSubnearAdditiveMagmaWithZero" Arg='A'/>
##
## <Description>
## is the additive magma-with-zero that has the zero of
## the near-additive magma-with-zero <A>A</A> as its only element.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "TrivialSubnearAdditiveMagmaWithZero",
IsNearAdditiveMagmaWithZero );
DeclareSynonymAttr( "TrivialSubadditiveMagmaWithZero",
TrivialSubnearAdditiveMagmaWithZero );
#############################################################################
##
#A AdditiveNeutralElement( <A> )
##
## <#GAPDoc Label="AdditiveNeutralElement">
## <ManSection>
## <Attr Name="AdditiveNeutralElement" Arg='A'/>
##
## <Description>
## returns the element <M>z</M> in the near-additive magma <A>A</A>
## with the property that <M>z + a = a = a + z</M> holds for all
## <M>a \in</M> <A>A</A>, if such an element exists.
## Otherwise <K>fail</K> is returned.
## <P/>
## A near-additive magma that is not a near-additive magma-with-zero
## can have an additive neutral element <M>z</M>;
## in this case, <M>z</M> <E>cannot</E> be obtained as
## <C>Zero( <A>A</A> )</C> or as <C>0*</C><M>a</M>
## for an element <M>a</M> in <A>A</A>, see <Ref Attr="Zero"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AdditiveNeutralElement", IsNearAdditiveMagma );
#############################################################################
##
#O ClosureNearAdditiveGroup( <A>, <a> ) . . for near-add. group and element
#O ClosureNearAdditiveGroup( <A>, <B> ) . . . . . for two near-add. groups
##
## <#GAPDoc Label="ClosureNearAdditiveGroup">
## <ManSection>
## <Heading>ClosureNearAdditiveGroup</Heading>
## <Oper Name="ClosureNearAdditiveGroup" Arg='A, a'
## Label="for a near-additive group and an element"/>
## <Oper Name="ClosureNearAdditiveGroup" Arg='A, B'
## Label="for two near-additive groups"/>
##
## <Description>
## returns the closure of the near-additive magma <A>A</A> with the element
## <A>a</A> or with the near-additive magma <A>B</A>, w.r.t. addition,
## taking the zero element, and taking additive inverses.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ClosureNearAdditiveGroup",
[ IsNearAdditiveGroup, IsNearAdditiveElement ] );
DeclareSynonym( "ClosureNearAdditiveMagmaWithInverses",
ClosureNearAdditiveGroup );
DeclareSynonym( "ClosureAdditiveGroup",
ClosureNearAdditiveGroup );
DeclareSynonym( "ClosureAdditiveMagmaWithInverses",
ClosureNearAdditiveGroup );
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