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############################################################################
##
## congruences/cong.gd
## Copyright (C) 2015-2022 Michael C. Young
##
## Licensing information can be found in the README file of this package.
##
############################################################################
##
## This file contains declarations for functions, operations and attributes of
## semigroup congruences that do not depend on any particular representation.
## Methods for most of these are implemented for specific types of congruence
## in the following files:
##
## congpart.gi - for congruences that can compute
## EquivalenceRelationPartition
## conginv.gi - Inverse semigroups
## congpairs.gi - Congruences with generating pairs
## congrees.gi - Rees congruences
## congrms.gi - (0-)simple Rees matrix semigroups
## congsimple.gi - (0-)simple semigroups
## conguniv.gi - Universal congruences
##
## Some general functions are also implemented in cong.gi, when these do not
## depend on the particular representation of the congruences, or that anything
## other than what is implemented in the GAP library is required. Most of the
## operations etc declared in this file are not implemented in cong.gi, but are
## declared here for congruences in general.
###############################################################################
# IsLeft/Right/SemigroupCongruence is a property, but
# IsLeft/Right/MagmaCongruence is a category so we use them in conjunction to
# mean i.e. a left congruence on a semigroup that was created in the category
# of left congruences. We introduce synonyms for these to simplify their use.
#
# Note that IsMagmaCongruence implies IsLeftMagmaCongruence and
# IsRightMagmaCongruence, and so IsLeftMagmaCongruence and
# IsLeftSemigroupCongruence returns true when applied to a 2-sided congruence.
# In other words, we cannot use IsLeftMagmaCongruence to determine whether or
# not a congruence was created as a left congruence or not (we can use
# IsLeftMagmaCongruence and not IsRightMagmaCongruence).
###############################################################################
DeclareProperty("IsLeftRightOrTwoSidedCongruence",
IsLeftMagmaCongruence and IsLeftSemigroupCongruence);
DeclareProperty("IsLeftRightOrTwoSidedCongruence",
IsRightMagmaCongruence and IsRightSemigroupCongruence);
DeclareProperty("IsLeftRightOrTwoSidedCongruence",
IsMagmaCongruence and IsSemigroupCongruence);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruence,
IsLeftMagmaCongruence and IsLeftSemigroupCongruence);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruence,
IsRightMagmaCongruence and IsRightSemigroupCongruence);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruence,
IsMagmaCongruence and IsSemigroupCongruence);
# The next attributes allows us to recover the category/string from a
# left/right/2-sided congruence object
DeclareAttribute("CongruenceHandednessString",
IsLeftRightOrTwoSidedCongruence);
############################################################################
# We introduce the property IsLeftRightOrTwoSidedCongruenceClass in a similar
# vein to IsLeftRightOrTwoSidedCongruence. Since we declare
# IsLeftCongruenceClass and IsRightCongruenceClass we could define them to
# satisfy IsLeftRightOrTwoSidedCongruenceClass, but then we have to declare
# IsLeftRightOrTwoSidedCongruenceClass as a being a property of
# IsEquivalenceClass, which means we then have to fiddle more with ranks of
# methods.
############################################################################
# IsCongruenceClass is declared in gap/lib/mgmcong.gd:140
# but it does not include IsAttributeStoringRep
DeclareCategory("IsLeftCongruenceClass",
IsEquivalenceClass and IsAttributeStoringRep);
DeclareCategory("IsRightCongruenceClass",
IsEquivalenceClass and IsAttributeStoringRep);
DeclareProperty("IsLeftRightOrTwoSidedCongruenceClass",
IsLeftCongruenceClass);
DeclareProperty("IsLeftRightOrTwoSidedCongruenceClass",
IsRightCongruenceClass);
DeclareProperty("IsLeftRightOrTwoSidedCongruenceClass",
IsCongruenceClass);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruenceClass,
IsLeftCongruenceClass);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruenceClass,
IsRightCongruenceClass);
InstallTrueMethod(IsLeftRightOrTwoSidedCongruenceClass,
IsCongruenceClass);
########################################################################
# Congruences
########################################################################
# Flexible functions for creating congruences
DeclareGlobalFunction("SemigroupCongruence");
DeclareGlobalFunction("LeftSemigroupCongruence");
DeclareGlobalFunction("RightSemigroupCongruence");
DeclareAttribute("TrivialCongruence", IsSemigroup);
# Properties of congruences
DeclareProperty("IsRightSemigroupCongruence", IsLeftSemigroupCongruence);
DeclareProperty("IsSemigroupCongruence", IsLeftSemigroupCongruence);
DeclareAttribute("NrEquivalenceClasses",
IsEquivalenceRelation);
DeclareAttribute("NonTrivialEquivalenceClasses",
IsEquivalenceRelation);
DeclareAttribute("EquivalenceRelationLookup",
IsEquivalenceRelation);
DeclareAttribute("EquivalenceRelationCanonicalLookup",
IsEquivalenceRelation);
DeclareAttribute("EquivalenceRelationCanonicalPartition",
IsEquivalenceRelation);
DeclareAttribute("EquivalenceRelationPartitionWithSingletons",
IsEquivalenceRelation);
# No-checks version of the "\in" operation
DeclareOperation("CongruenceTestMembershipNC",
[IsLeftRightOrTwoSidedCongruence,
IsMultiplicativeElement,
IsMultiplicativeElement]);
# Algebraic operators
DeclareOperation("JoinLeftSemigroupCongruences",
[IsLeftSemigroupCongruence, IsLeftSemigroupCongruence]);
DeclareOperation("JoinRightSemigroupCongruences",
[IsRightSemigroupCongruence, IsRightSemigroupCongruence]);
DeclareOperation("MeetLeftSemigroupCongruences",
[IsLeftSemigroupCongruence, IsLeftSemigroupCongruence]);
DeclareOperation("MeetRightSemigroupCongruences",
[IsRightSemigroupCongruence, IsRightSemigroupCongruence]);
# Comparison operators
DeclareOperation("IsSubrelation",
[IsEquivalenceRelation, IsEquivalenceRelation]);
DeclareOperation("IsSuperrelation",
[IsEquivalenceRelation, IsEquivalenceRelation]);
########################################################################
# Congruence classes
########################################################################
# Actions
DeclareOperation("OnLeftCongruenceClasses",
[IsLeftRightOrTwoSidedCongruenceClass,
IsMultiplicativeElement]);
DeclareOperation("OnRightCongruenceClasses",
[IsLeftRightOrTwoSidedCongruenceClass,
IsMultiplicativeElement]);
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