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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 semirings.
##
#############################################################################
##
#P IsLDistributive( <C> )
##
## <#GAPDoc Label="IsLDistributive">
## <ManSection>
## <Prop Name="IsLDistributive" Arg='C'/>
##
## <Description>
## is <K>true</K> if the relation
## <M>a * ( b + c ) = ( a * b ) + ( a * c )</M>
## holds for all elements <M>a</M>, <M>b</M>, <M>c</M> in the collection
## <A>C</A>, and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsLDistributive", IsRingElementCollection );
InstallSubsetMaintenance( IsLDistributive,
IsRingElementCollection and IsLDistributive,
IsRingElementCollection );
InstallFactorMaintenance( IsLDistributive,
IsRingElementCollection and IsLDistributive,
IsObject,
IsRingElementCollection );
#############################################################################
##
#P IsRDistributive( <C> )
##
## <#GAPDoc Label="IsRDistributive">
## <ManSection>
## <Prop Name="IsRDistributive" Arg='C'/>
##
## <Description>
## is <K>true</K> if the relation
## <M>( a + b ) * c = ( a * c ) + ( b * c )</M>
## holds for all elements <M>a</M>, <M>b</M>, <M>c</M> in the collection
## <A>C</A>, and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsRDistributive", IsRingElementCollection );
InstallSubsetMaintenance( IsRDistributive,
IsRingElementCollection and IsRDistributive,
IsRingElementCollection );
InstallFactorMaintenance( IsRDistributive,
IsRingElementCollection and IsRDistributive,
IsObject,
IsRingElementCollection );
#############################################################################
##
#P IsDistributive( <C> )
##
## <#GAPDoc Label="IsDistributive">
## <ManSection>
## <Prop Name="IsDistributive" Arg='C'/>
##
## <Description>
## is <K>true</K> if the collection <A>C</A> is both left and right
## distributive
## (see <Ref Prop="IsLDistributive"/>, <Ref Prop="IsRDistributive"/>),
## and <K>false</K> otherwise.
## <Example><![CDATA[
## gap> IsDistributive( Integers );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonymAttr( "IsDistributive", IsLDistributive and IsRDistributive );
#############################################################################
##
#P IsSemiring( <S> )
##
## <ManSection>
## <Prop Name="IsSemiring" Arg='S'/>
##
## <Description>
## A <E>semiring</E> in &GAP; is an additive magma (see <Ref Func="IsAdditiveMagma"/>)
## that is also a magma (see <Ref Func="IsMagma"/>),
## such that addition <C>+</C> and multiplication <C>*</C> are distributive.
## <P/>
## The multiplication need <E>not</E> be associative (see <Ref Func="IsAssociative"/>).
## For example, a Lie algebra (see <Ref Chap="Lie Algebras"/>) is regarded as a
## semiring in &GAP;.
## A semiring need not have an identity and a zero element,
## see <Ref Prop="IsSemiringWithOne"/> and <Ref Prop="IsSemiringWithZero"/>.
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsSemiring",
IsAdditiveMagma and IsMagma and IsDistributive );
#############################################################################
##
#P IsSemiringWithOne( <S> )
##
## <ManSection>
## <Prop Name="IsSemiringWithOne" Arg='S'/>
##
## <Description>
## A <E>semiring-with-one</E> in &GAP; is a semiring (see <Ref Prop="IsSemiring"/>)
## that is also a magma-with-one (see <Ref Func="IsMagmaWithOne"/>).
## <P/>
## Note that a semiring-with-one need not contain a zero element
## (see <Ref Prop="IsSemiringWithZero"/>).
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsSemiringWithOne",
IsAdditiveMagma and IsMagmaWithOne and IsDistributive );
#############################################################################
##
#P IsSemiringWithZero( <S> )
##
## <ManSection>
## <Prop Name="IsSemiringWithZero" Arg='S'/>
##
## <Description>
## A <E>semiring-with-zero</E> in &GAP; is a semiring (see <Ref Prop="IsSemiring"/>)
## that is also an additive magma-with-zero (see <Ref Func="IsAdditiveMagmaWithZero"/>).
## <P/>
## Note that a semiring-with-zero need not contain an identity element
## (see <Ref Prop="IsSemiringWithOne"/>).
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsSemiringWithZero",
IsAdditiveMagmaWithZero and IsMagma and IsDistributive );
#############################################################################
##
#P IsSemiringWithOneAndZero( <S> )
##
## <ManSection>
## <Prop Name="IsSemiringWithOneAndZero" Arg='S'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareSynonymAttr( "IsSemiringWithOneAndZero",
IsAdditiveMagmaWithZero and IsMagmaWithOne and IsDistributive );
#############################################################################
##
#A GeneratorsOfSemiring( <S> )
##
## <ManSection>
## <Attr Name="GeneratorsOfSemiring" Arg='S'/>
##
## <Description>
## <C>GeneratorsOfSemiring</C> returns a list of elements such that
## the semiring <A>S</A> is the closure of these elements
## under addition and multiplication.
## </Description>
## </ManSection>
##
DeclareAttribute( "GeneratorsOfSemiring", IsSemiring );
#############################################################################
##
#A GeneratorsOfSemiringWithOne( <S> )
##
## <ManSection>
## <Attr Name="GeneratorsOfSemiringWithOne" Arg='S'/>
##
## <Description>
## <C>GeneratorsOfSemiringWithOne</C> returns a list of elements such that
## the semiring <A>R</A> is the closure of these elements
## under addition, multiplication, and taking the identity element
## <C>One( <A>S</A> )</C>.
## <P/>
## <A>S</A> itself need <E>not</E> be known to be a semiring-with-one.
## </Description>
## </ManSection>
##
DeclareAttribute( "GeneratorsOfSemiringWithOne", IsSemiringWithOne );
#############################################################################
##
#A GeneratorsOfSemiringWithZero( <S> )
##
## <ManSection>
## <Attr Name="GeneratorsOfSemiringWithZero" Arg='S'/>
##
## <Description>
## <C>GeneratorsOfSemiringWithZero</C> returns a list of elements such that
## the semiring <A>S</A> is the closure of these elements
## under addition, multiplication, and taking the zero element
## <C>Zero( <A>S</A> )</C>.
## <P/>
## <A>S</A> itself need <E>not</E> be known to be a semiring-with-zero.
## </Description>
## </ManSection>
##
DeclareAttribute( "GeneratorsOfSemiringWithZero", IsSemiringWithZero );
#############################################################################
##
#A GeneratorsOfSemiringWithOneAndZero( <S> )
##
## <ManSection>
## <Attr Name="GeneratorsOfSemiringWithOneAndZero" Arg='S'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "GeneratorsOfSemiringWithOneAndZero",
IsSemiringWithOneAndZero );
#############################################################################
##
#A AsSemiring( <C> )
##
## <ManSection>
## <Attr Name="AsSemiring" Arg='C'/>
##
## <Description>
## If the elements in the collection <A>C</A> form a semiring
## then <C>AsSemiring</C> returns this semiring,
## otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
##
DeclareAttribute( "AsSemiring", IsRingElementCollection );
#############################################################################
##
#A AsSemiringWithOne( <C> )
##
## <ManSection>
## <Attr Name="AsSemiringWithOne" Arg='C'/>
##
## <Description>
## If the elements in the collection <A>C</A> form a semiring-with-one
## then <C>AsSemiringWithOne</C> returns this semiring-with-one,
## otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
##
DeclareAttribute( "AsSemiringWithOne", IsRingElementCollection );
#############################################################################
##
#A AsSemiringWithZero( <C> )
##
## <ManSection>
## <Attr Name="AsSemiringWithZero" Arg='C'/>
##
## <Description>
## If the elements in the collection <A>C</A> form a semiring-with-zero
## then <C>AsSemiringWithZero</C> returns this semiring-with-zero,
## otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
##
DeclareAttribute( "AsSemiringWithZero", IsRingElementCollection );
#############################################################################
##
#A AsSemiringWithOneAndZero( <C> )
##
## <ManSection>
## <Attr Name="AsSemiringWithOneAndZero" Arg='C'/>
##
## <Description>
## If the elements in the collection <A>C</A> form a semiring-with-one-and-zero
## then <C>AsSemiringWithOneAndZero</C> returns this semiring-with-one-and-zero,
## otherwise <K>fail</K> is returned.
## </Description>
## </ManSection>
##
DeclareAttribute( "AsSemiringWithOneAndZero", IsRingElementCollection );
#############################################################################
##
#O ClosureSemiring( <S>, <s> )
#O ClosureSemiring( <S>, <T> )
##
## <ManSection>
## <Oper Name="ClosureSemiring" Arg='S, s'/>
## <Oper Name="ClosureSemiring" Arg='S, T'/>
##
## <Description>
## For a semiring <A>S</A> and either an element <A>s</A> of its elements family
## or a semiring <A>T</A>,
## <C>ClosureSemiring</C> returns the semiring generated by both arguments.
## </Description>
## </ManSection>
##
DeclareOperation( "ClosureSemiring", [ IsSemiring, IsObject ] );
#############################################################################
##
#O SemiringByGenerators( <C> ) . . . semiring gener. by elements in a coll.
##
## <ManSection>
## <Oper Name="SemiringByGenerators" Arg='C'/>
##
## <Description>
## <C>SemiringByGenerators</C> returns the semiring generated by the elements
## in the collection <A>C</A>,
## i. e., the closure of <A>C</A> under addition and multiplication.
## </Description>
## </ManSection>
##
DeclareOperation( "SemiringByGenerators", [ IsCollection ] );
#############################################################################
##
#O SemiringWithOneByGenerators( <C> )
##
## <ManSection>
## <Oper Name="SemiringWithOneByGenerators" Arg='C'/>
##
## <Description>
## <C>SemiringWithOneByGenerators</C> returns the semiring-with-one generated by
## the elements in the collection <A>C</A>, i. e., the closure of <A>C</A> under
## addition, multiplication, and taking the identity of an element.
## </Description>
## </ManSection>
##
DeclareOperation( "SemiringWithOneByGenerators", [ IsCollection ] );
#############################################################################
##
#O SemiringWithZeroByGenerators( <C> )
##
## <ManSection>
## <Oper Name="SemiringWithZeroByGenerators" Arg='C'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareOperation( "SemiringWithZeroByGenerators", [ IsCollection ] );
#############################################################################
##
#O SemiringWithOneAndZeroByGenerators( <C> )
##
## <ManSection>
## <Oper Name="SemiringWithOneAndZeroByGenerators" Arg='C'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareOperation( "SemiringWithOneAndZeroByGenerators", [ IsCollection ] );
#############################################################################
##
#F Semiring( <r> ,<s>, ... ) . . . . . . semiring generated by a collection
#F Semiring( <C> ) . . . . . . . . . . . semiring generated by a collection
##
## <ManSection>
## <Func Name="Semiring" Arg='r ,s, ...'/>
## <Func Name="Semiring" Arg='C'/>
##
## <Description>
## In the first form <C>Semiring</C> returns the smallest semiring that
## contains all the elements <A>r</A>, <A>s</A>... etc.
## In the second form <C>Semiring</C> returns the smallest semiring that
## contains all the elements in the collection <A>C</A>.
## If any element is not an element of a semiring or if the elements lie in
## no common semiring an error is raised.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "Semiring" );
#############################################################################
##
#F SemiringWithOne( <r>, <s>, ... )
#F SemiringWithOne( <C> )
##
## <ManSection>
## <Func Name="SemiringWithOne" Arg='r, s, ...'/>
## <Func Name="SemiringWithOne" Arg='C'/>
##
## <Description>
## In the first form <C>SemiringWithOne</C> returns the smallest
## semiring-with-one that contains all the elements <A>r</A>, <A>s</A>... etc.
## In the second form <C>SemiringWithOne</C> returns the smallest
## semiring-with-one that contains all the elements in the collection <A>C</A>.
## If any element is not an element of a semiring or if the elements lie in
## no common semiring an error is raised.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SemiringWithOne" );
#############################################################################
##
#F SemiringWithZero( <r>, <s>, ... )
#F SemiringWithZero( <C> )
##
## <ManSection>
## <Func Name="SemiringWithZero" Arg='r, s, ...'/>
## <Func Name="SemiringWithZero" Arg='C'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SemiringWithZero" );
#############################################################################
##
#F SemiringWithOneAndZero( <r>, <s>, ... )
#F SemiringWithOneAndZero( <C> )
##
## <ManSection>
## <Func Name="SemiringWithOneAndZero" Arg='r, s, ...'/>
## <Func Name="SemiringWithOneAndZero" Arg='C'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SemiringWithOneAndZero" );
#############################################################################
##
#F Subsemiring( <S>, <gens> )
#F SubsemiringNC( <S>, <gens> )
##
## <ManSection>
## <Func Name="Subsemiring" Arg='S, gens'/>
## <Func Name="SubsemiringNC" Arg='S, gens'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "Subsemiring" );
DeclareGlobalFunction( "SubsemiringNC" );
#############################################################################
##
#F SubsemiringWithOne( <S>, <gens> )
#F SubsemiringWithOneNC( <S>, <gens> )
##
## <ManSection>
## <Func Name="SubsemiringWithOne" Arg='S, gens'/>
## <Func Name="SubsemiringWithOneNC" Arg='S, gens'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SubsemiringWithOne" );
DeclareGlobalFunction( "SubsemiringWithOneNC" );
#############################################################################
##
#F SubsemiringWithZero( <S>, <gens> )
#F SubsemiringWithZeroNC( <S>, <gens> )
##
## <ManSection>
## <Func Name="SubsemiringWithZero" Arg='S, gens'/>
## <Func Name="SubsemiringWithZeroNC" Arg='S, gens'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SubsemiringWithZero" );
DeclareGlobalFunction( "SubsemiringWithZeroNC" );
#############################################################################
##
#F SubsemiringWithOneAndZero( <S>, <gens> )
#F SubsemiringWithOneAndZeroNC( <S>, <gens> )
##
## <ManSection>
## <Func Name="SubsemiringWithOneAndZero" Arg='S, gens'/>
## <Func Name="SubsemiringWithOneAndZeroNC" Arg='S, gens'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SubsemiringWithOneAndZero" );
DeclareGlobalFunction( "SubsemiringWithOneAndZeroNC" );
#############################################################################
##
#A CentralIdempotentsOfSemiring( <S> )
##
## <ManSection>
## <Attr Name="CentralIdempotentsOfSemiring" Arg='S'/>
##
## <Description>
## For a semiring <A>S</A>, this function returns
## a list of central primitive idempotents such that their sum is
## the identity element of <A>S</A>.
## Therefore <A>S</A> is required to have an identity.
## </Description>
## </ManSection>
##
DeclareAttribute( "CentralIdempotentsOfSemiring", IsSemiring );
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