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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 implementation for quotient semigroups.
##
#############################################################################
##
#F HomomorphismQuotientSemigroup(<cong>)
##
##
InstallGlobalFunction(HomomorphismQuotientSemigroup,
function(cong)
local S, Qrep, efam, filters, Q, Qgens;
if not IsSemigroupCongruence(cong) then
ErrorNoReturn("usage: the argument should be a semigroup congruence,");
fi;
S := Source(cong);
Qrep := EquivalenceClassOfElementNC(cong, Representative(S));
efam := FamilyObj(Qrep);
filters := IsSemigroup and IsQuotientSemigroup and IsAttributeStoringRep;
if IsMonoid(S) then
filters := filters and IsMagmaWithOne;
fi;
if HasIsFinite(S) and IsFinite(S) then
filters := filters and IsFinite;
fi;
Q:=Objectify(NewType( CollectionsFamily( efam ), filters), rec() );
SetRepresentative(Q, Qrep);
SetQuotientSemigroupPreimage(Q, S);
SetQuotientSemigroupCongruence(Q, cong);
SetQuotientSemigroupHomomorphism(Q, MagmaHomomorphismByFunctionNC(S, Q,
x->EquivalenceClassOfElementNC(cong,x)));
efam!.quotient := Q;
if IsMonoid(Q) and HasOne(S) then
SetOne(Q, One(S)^QuotientSemigroupHomomorphism(Q));
fi;
# Set the semigroup generators of the quotient if possible.
#
# It is not sufficient to check HasGeneratorsOfSemigroup
# by itself, because groups for example only have
# GeneratorsOfMagmaWithInverses.
#
# The code below is safe, because GeneratorsOfSemigroup
# is synonymous with GeneratorsOfMagma, and there is a
# method for GeneratorsOfMagma, if we know
# GeneratorsOfMagmaWithInverses.
if HasGeneratorsOfMagma(S) or
HasGeneratorsOfMagmaWithInverses(S) or
HasGeneratorsOfSemigroup(S) then
Qgens := List( GeneratorsOfSemigroup(S),
s -> s^QuotientSemigroupHomomorphism(Q));
SetGeneratorsOfSemigroup( Q, Qgens );
fi;
return QuotientSemigroupHomomorphism(Q);
end);
#############################################################################
##
#M HomomorphismFactorSemigroup(<s>, <cong> )
##
## for a generic semigroup and congruence
##
InstallMethod(HomomorphismFactorSemigroup,
"for a semigroup and a congruence",
true,
[ IsSemigroup, IsSemigroupCongruence ],
0,
function(s, c)
if not s = Source(c) then
TryNextMethod();
fi;
return HomomorphismQuotientSemigroup(c);
end);
#############################################################################
##
#M ViewObj( S )
##
## View a quotient semigroup S
##
InstallMethod( ViewObj,
"for a quotient semigroup with generators",
true,
[ IsQuotientSemigroup], 0,
function( S )
Print( "<quotient of ",QuotientSemigroupPreimage(S)," by ",
QuotientSemigroupCongruence(S),">");
end );
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