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############################################################################
##
## congruences/congrees.gi
## Copyright (C) 2015-2022 Michael C. Young
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
## This file contains methods for Rees congruences; i.e. semigroup congruences
## defined by a two-sided ideal. See Howie 1.7
##
#############################################################################
# Testers + fundamental representation specific attributes
#############################################################################
# The following requires CanComputeEquivalenceRelationPartition, because
# otherwise, it applies to other congruences, which don't have a method for
# NrEquivalenceClasses.
InstallMethod(IsReesCongruence, "for a semigroup congruence",
[CanComputeEquivalenceRelationPartition],
9, # to beat the library method
function(C)
local S, classes, nontrivial, i, class, I;
if not IsSemigroupCongruence(C) then
return false;
fi;
# This function is adapted from code in the library
S := Range(C);
if NrEquivalenceClasses(C) = Size(S) then
# Trivial congruence - only possible I is zero
if MultiplicativeZero(S) <> fail then
SetSemigroupIdealOfReesCongruence(C, MinimalIdeal(S));
return true;
else
return false;
fi;
else
# Search for non-trivial classes
classes := EquivalenceClasses(C);
nontrivial := 0;
for i in [1 .. Length(classes)] do
if Size(classes[i]) > 1 then
if nontrivial = 0 then
nontrivial := i;
else
# Two non-trivial classes
return false;
fi;
fi;
od;
# Only one non-trivial class - check it is an I
class := classes[nontrivial];
I := SemigroupIdeal(S, AsList(class));
if Size(class) = Size(I) then
SetSemigroupIdealOfReesCongruence(C, I);
return true;
else
return false;
fi;
fi;
end);
InstallMethod(ReesCongruenceOfSemigroupIdeal, "for a semigroup ideal",
[IsSemigroupIdeal],
function(I)
local S, fam, type, C;
S := Parent(I);
# Construct the object
fam := GeneralMappingsFamily(ElementsFamily(FamilyObj(S)),
ElementsFamily(FamilyObj(S)));
type := NewType(fam,
IsSemigroupCongruence
and IsMagmaCongruence
and IsAttributeStoringRep
and CanComputeEquivalenceRelationPartition);
C := Objectify(type, rec());
# Set some attributes
SetSource(C, S);
SetRange(C, S);
SetSemigroupIdealOfReesCongruence(C, I);
SetIsReesCongruence(C, true);
return C;
end);
InstallMethod(AsSemigroupCongruenceByGeneratingPairs, "for a Rees congruence",
[IsReesCongruence],
function(C)
local S, gens, min, pairs, y, x;
S := Range(C);
gens := MinimalIdealGeneratingSet(SemigroupIdealOfReesCongruence(C));
min := MinimalIdeal(S);
pairs := [];
C := SemigroupCongruence(S, pairs);
for y in min do
for x in gens do
if not [x, y] in C then
Add(pairs, [x, y]);
C := SemigroupCongruence(S, pairs);
fi;
od;
od;
return C;
end);
InstallMethod(GeneratingPairsOfMagmaCongruence, "for a Rees congruence",
[IsReesCongruence],
function(C)
C := AsSemigroupCongruenceByGeneratingPairs(C);
return GeneratingPairsOfSemigroupCongruence(C);
end);
InstallMethod(ViewObj, "for a Rees congruence",
[IsReesCongruence],
function(C)
Print("<Rees congruence of ");
ViewObj(SemigroupIdealOfReesCongruence(C));
Print(" over ");
ViewObj(Range(C));
Print(">");
end);
InstallMethod(PrintObj, "for a Rees congruence",
[IsReesCongruence],
function(C)
Print("ReesCongruenceOfSemigroupIdeal( ");
Print(SemigroupIdealOfReesCongruence(C));
Print(" )");
end);
#############################################################################
# The mandatory things that must be implemented for
# congruences belonging to CanComputeEquivalenceRelationPartition
#############################################################################
InstallMethod(EquivalenceRelationPartition, "for a Rees congruence",
[IsReesCongruence], C -> [AsList(SemigroupIdealOfReesCongruence(C))]);
InstallMethod(ImagesElm,
"for a Rees congruence and a multiplicative element",
[IsReesCongruence, IsMultiplicativeElement],
function(C, x)
if not x in Range(C) then
ErrorNoReturn("the 2nd argument (a mult. elt.) does not belong to ",
"the range of the 1st argument (a Rees congruence)");
elif x in SemigroupIdealOfReesCongruence(C) then
return Elements(SemigroupIdealOfReesCongruence(C));
else
return [x];
fi;
end);
InstallMethod(CongruenceTestMembershipNC,
"for Rees congruence and two multiplicative elements",
[IsReesCongruence, IsMultiplicativeElement, IsMultiplicativeElement],
function(C, lhop, rhop)
local I;
I := SemigroupIdealOfReesCongruence(C);
return (lhop = rhop) or (lhop in I and rhop in I);
end);
########################################################################
# The non-mandatory things where we have a superior method for this particular
# representation.
########################################################################
########################################################################
# Operators
########################################################################
InstallMethod(\=, "for two Rees congruences",
[IsReesCongruence, IsReesCongruence],
function(lhop, rhop)
return SemigroupIdealOfReesCongruence(lhop) =
SemigroupIdealOfReesCongruence(rhop);
end);
InstallMethod(IsSubrelation, "for two Rees congruences",
[IsReesCongruence, IsReesCongruence],
function(lhop, rhop)
local I1, I2;
# Tests whether rhop is a subcongruence of lhop
if Range(lhop) <> Range(rhop) then
Error("the 1st and 2nd arguments are congruences over different",
" semigroups");
fi;
I1 := SemigroupIdealOfReesCongruence(lhop);
I2 := SemigroupIdealOfReesCongruence(rhop);
return ForAll(GeneratorsOfSemigroupIdeal(I2), gen -> gen in I1);
end);
InstallMethod(JoinSemigroupCongruences, "for two Rees congruences",
[IsReesCongruence, IsReesCongruence],
function(lhop, rhop)
local gens1, gens2, I;
if Range(lhop) <> Range(rhop) then
Error("cannot form the join of congruences over different semigroups");
fi;
gens1 := GeneratorsOfSemigroupIdeal(SemigroupIdealOfReesCongruence(lhop));
gens2 := GeneratorsOfSemigroupIdeal(SemigroupIdealOfReesCongruence(rhop));
I := SemigroupIdeal(Range(lhop), Concatenation(gens1, gens2));
I := SemigroupIdeal(Range(lhop), MinimalIdealGeneratingSet(I));
return ReesCongruenceOfSemigroupIdeal(I);
end);
########################################################################
# Equivalence classes
########################################################################
InstallMethod(NrEquivalenceClasses, "for a Rees congruence",
[IsReesCongruence],
C -> Size(Range(C)) - Size(SemigroupIdealOfReesCongruence(C)) + 1);
InstallMethod(EquivalenceClasses, "for a Rees congruence", [IsReesCongruence],
function(C)
local classes, I, next, x;
classes := EmptyPlist(NrEquivalenceClasses(C));
I := SemigroupIdealOfReesCongruence(C);
classes[1] := EquivalenceClassOfElementNC(C, I.1);
next := 2;
for x in Range(C) do
if not (x in I) then
classes[next] := EquivalenceClassOfElementNC(C, x);
next := next + 1;
fi;
od;
return classes;
end);
InstallMethod(NonTrivialEquivalenceClasses, "for a Rees congruence",
[IsReesCongruence],
function(C)
local I;
I := SemigroupIdealOfReesCongruence(C);
if Size(I) = 1 then
return [];
fi;
return [EquivalenceClassOfElementNC(C, I.1)];
end);
InstallMethod(EquivalenceClassOfElementNC,
"for a Rees congruence and a multiplicative element",
[IsReesCongruence, IsMultiplicativeElement],
function(C, x)
local is_ideal_class, fam, class;
# Ensure consistency of representatives
if x in SemigroupIdealOfReesCongruence(C) then
is_ideal_class := true;
x := SemigroupIdealOfReesCongruence(C).1;
else
is_ideal_class := false;
fi;
# Construct the object
fam := CollectionsFamily(FamilyObj(x));
class := Objectify(NewType(fam,
IsReesCongruenceClass
and IsLeftRightOrTwoSidedCongruenceClass),
rec());
if is_ideal_class then
SetSize(class, Size(SemigroupIdealOfReesCongruence(C)));
else
SetSize(class, 1);
fi;
SetParentAttr(class, Range(C));
SetEquivalenceClassRelation(class, C);
SetRepresentative(class, x);
return class;
end);
########################################################################
# Operators for classes
########################################################################
InstallMethod(\*, "for two Rees congruence classes",
[IsReesCongruenceClass, IsReesCongruenceClass],
function(lhop, rhop)
if EquivalenceClassRelation(lhop) <> EquivalenceClassRelation(rhop) then
ErrorNoReturn("the arguments (cong. classes) are not classes of the same ",
"congruence");
elif Size(lhop) > 1 then
return lhop;
elif Size(rhop) > 1 then
return rhop;
fi;
return EquivalenceClassOfElementNC(EquivalenceClassRelation(lhop),
Representative(lhop) *
Representative(rhop));
end);
InstallMethod(\=, "for two Rees congruence classes",
[IsReesCongruenceClass, IsReesCongruenceClass],
function(lhop, rhop)
return EquivalenceClassRelation(lhop) = EquivalenceClassRelation(rhop)
and (Representative(lhop) = Representative(rhop))
or (Size(lhop) > 1 and Size(rhop) > 1);
end);
InstallMethod(\in,
"for a multiplicative element and a Rees congruence class",
[IsMultiplicativeElement, IsReesCongruenceClass],
function(x, class)
local rel;
if Size(class) > 1 then
rel := EquivalenceClassRelation(class);
return x in SemigroupIdealOfReesCongruence(rel);
else
return x = Representative(class);
fi;
end);
[ Dauer der Verarbeitung: 0.26 Sekunden
(vorverarbeitet)
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