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##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Andrew Solomon and Isabel Araújo.
##
## 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 declarations for quotient semigroups.
##
## <#GAPDoc Label="[1]{semiquo}">
## For a semigroup <M>S</M>,
## elements of a quotient semigroup are equivalence classes of
## elements of the <Ref Attr="QuotientSemigroupPreimage"/> value
## under the congruence given by the value of
## <Ref Attr="QuotientSemigroupCongruence"/>.
## <P/>
## It is probably most useful for calculating the elements of
## the equivalence classes by using <Ref Func="Elements"/> or by looking at
## the images of elements of <Ref Attr="QuotientSemigroupPreimage"/> under
## the map returned by <Ref Attr="QuotientSemigroupHomomorphism"/>,
## which maps the <Ref Attr="QuotientSemigroupPreimage"/> value to <A>S</A>.
## <P/>
## For intensive computations in a quotient semigroup, it is probably
## worthwhile finding another representation as the equality test
## could involve enumeration of the elements of the congruence classes
## being compared.
## <#/GAPDoc>
##
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##
#C IsQuotientSemigroup( <S> )
##
## <#GAPDoc Label="IsQuotientSemigroup">
## <ManSection>
## <Filt Name="IsQuotientSemigroup" Arg='S' Type='Category'/>
##
## <Description>
## is the category of semigroups constructed from another semigroup
## and a congruence on it.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory("IsQuotientSemigroup", IsSemigroup);
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##
#F HomomorphismQuotientSemigroup(<cong>)
##
## <#GAPDoc Label="HomomorphismQuotientSemigroup">
## <ManSection>
## <Func Name="HomomorphismQuotientSemigroup" Arg='cong'/>
##
## <Description>
## for a congruence <A>cong</A> and a semigroup <A>S</A>.
## Returns the homomorphism from <A>S</A> to the quotient of <A>S</A>
## by <A>cong</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("HomomorphismQuotientSemigroup");
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##
#A QuotientSemigroupPreimage(<S>)
#A QuotientSemigroupCongruence(<S>)
#A QuotientSemigroupHomomorphism(<S>)
##
## <#GAPDoc Label="QuotientSemigroupPreimage">
## <ManSection>
## <Attr Name="QuotientSemigroupPreimage" Arg='S'/>
## <Attr Name="QuotientSemigroupCongruence" Arg='S'/>
## <Attr Name="QuotientSemigroupHomomorphism" Arg='S'/>
##
## <Description>
## for a quotient semigroup <A>S</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("QuotientSemigroupPreimage", IsQuotientSemigroup);
DeclareAttribute("QuotientSemigroupCongruence", IsQuotientSemigroup);
DeclareAttribute("QuotientSemigroupHomomorphism", IsQuotientSemigroup);
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