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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Andrew Solomon.
##
## 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 generic methods for semigroup congruences.
##
## Maintenance and further development by:
## Robert F. Morse
## Andrew Solomon
##
######################################################################
##
##
#P LeftSemigroupCongruenceByGeneratingPairs( <semigroup>, <gens> )
#P RightSemigroupCongruenceByGeneratingPairs( <semigroup>, <gens> )
#P SemigroupCongruenceByGeneratingPairs( <semigroup>, <gens> )
##
##
######################################################################
InstallMethod( LeftSemigroupCongruenceByGeneratingPairs,
"for a Semigroup and a list of pairs of its elements",
IsElmsColls,
[ IsSemigroup, IsList ], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsLeftMagmaCongruence);
SetIsLeftSemigroupCongruence(cong,true);
return cong;
end );
InstallMethod( LeftSemigroupCongruenceByGeneratingPairs,
"for a Semigroup and an empty list",
true,
[ IsSemigroup, IsList and IsEmpty ], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsLeftMagmaCongruence);
SetIsLeftSemigroupCongruence(cong,true);
SetEquivalenceRelationPartition(cong,[]);
return cong;
end );
InstallMethod( RightSemigroupCongruenceByGeneratingPairs,
"for a Semigroup and a list of pairs of its elements",
IsElmsColls,
[ IsSemigroup, IsList ], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsRightMagmaCongruence);
SetIsRightSemigroupCongruence(cong,true);
return cong;
end );
InstallMethod( RightSemigroupCongruenceByGeneratingPairs,
"for a Semigroup and an empty list",
true,
[ IsSemigroup, IsList and IsEmpty ], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsRightMagmaCongruence);
SetIsRightSemigroupCongruence(cong,true);
SetEquivalenceRelationPartition(cong,[]);
return cong;
end );
InstallMethod( SemigroupCongruenceByGeneratingPairs,
"for a semigroup and a list of pairs of its elements",
IsElmsColls,
[ IsSemigroup, IsList ], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsMagmaCongruence);
SetIsSemigroupCongruence(cong,true);
return cong;
end );
InstallMethod( SemigroupCongruenceByGeneratingPairs,
"for a semigroup and an empty list",
true,
[ IsSemigroup, IsList and IsEmpty], 0,
function( M, gens )
local cong;
cong := LR2MagmaCongruenceByGeneratingPairsCAT(M, gens,
IsMagmaCongruence);
SetIsSemigroupCongruence(cong,true);
SetEquivalenceRelationPartition(cong,[]);
return cong;
end );
#############################################################################
##
#P IsLeftSemigroupCongruence(<c>)
#P IsRightSemigroupCongruence(<c>)
#P IsSemigroupCongruence(<c>)
##
InstallMethod( IsLeftSemigroupCongruence,
"test whether a left magma congruence is a semigroup a congruence",
true,
[ IsLeftMagmaCongruence ], 0,
function(c)
return IsSemigroup(Source(c));
end);
InstallMethod( IsRightSemigroupCongruence,
"test whether a right magma congruence is a semigroup a congruence",
true,
[ IsRightMagmaCongruence ], 0,
function(c)
return IsSemigroup(Source(c));
end);
InstallMethod( IsSemigroupCongruence,
"test whether a magma congruence is a semigroup a congruence",
true,
[ IsMagmaCongruence ], 0,
function(c)
return IsSemigroup(Source(c));
end);
#############################################################################
##
#M IsReesCongruence(<c>)
##
## True when the congruence has at most one
## nonsingleton congruence class and that equivalence
## class forms an ideal of the semigroup.
## A special check is needed if the congruence is the
## diagonal congruence -- as this congruence is a Rees
## congruence only if the semigroup contains a zero element.
##
InstallMethod( IsReesCongruence,
"for a semigroup congruence",
true,
[ IsSemigroupCongruence ], 0,
function( cong )
local part, # partition
id, # ideal generated by non singleton block
it, # iterator of id
s; # underlying semigroup
part := EquivalenceRelationPartition(cong);
# Determine if the congruence is Green's relation
# we have slightly different attributes as the
# relation is represented on points.
#
if IsGreensRelation(cong) then
s := AssociatedSemigroup(cong);
part := List(part, x->List(x,y->AsSortedList(s)[y]));
else
s := Source(cong);
fi;
if Length(part)=0 then
# if all blocks are singletons we must check to see
# if the semigroup contains a zero otherwise return false.
#
# See if it already has one -- if so return true
#
if HasMultiplicativeZero(s) then return true; fi;
# The semigroup might have a zero it just isn't identified
# yet.
#
# Using the IsMultiplicativeZero method for semigroups
# is the most efficient which only checks with the
# generators of the semigroup. We prune our search to the
# idempotents.
return ForAny(Idempotents(s), x->IsMultiplicativeZero(s,x));
elif Length(part)=1 then
# if there is one non singletion block
# check that it forms an ideal
id := MagmaIdealByGenerators(s,part[1]);
# loop through the elements of the ideal id
# until you find an element not in the non singleton block
it := Iterator(id);
while not IsDoneIterator(it) do
if not NextIterator(it) in part[1] then
return false;
fi;
od;
# here we know that the block forms an ideal
# hence the congruence is Rees
return true;
else
# if the partition has more than one non singleton class
# then it is not a Rees congruence
return false;
fi;
end);
#############################################################################
##
#M PrintObj( <smg cong> )
##
## left semigroup congruence
##
InstallMethod( PrintObj,
"for a left semigroup congruence",
true,
[ IsLeftSemigroupCongruence ], 0,
function( S )
Print( "LeftSemigroupCongruence( ... )" );
end );
InstallMethod( PrintObj,
"for a left semigroup congruence with known generating pairs",
true,
[ IsLeftSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "LeftSemigroupCongruence( ",
GeneratingPairsOfMagmaCongruence( S ), " )" );
end );
## right semigroup congruence
InstallMethod( PrintObj,
"for a right semigroup congruence",
true,
[ IsRightSemigroupCongruence ], 0,
function( S )
Print( "RightSemigroupCongruence( ... )" );
end );
InstallMethod( PrintObj,
"for a right semigroup congruence with known generating pairs",
true,
[ IsRightSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "RightSemigroupCongruence( ",
GeneratingPairsOfMagmaCongruence( S ), " )" );
end );
## two sided semigroup congruence
InstallMethod( PrintObj,
"for a semigroup congruence",
true,
[ IsSemigroupCongruence ], 0,
function( S )
Print( "SemigroupCongruence( ... )" );
end );
InstallMethod( PrintObj,
"for a semigroup Congruence with known generating pairs",
true,
[ IsSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "SemigroupCongruence( ",
GeneratingPairsOfMagmaCongruence( S ), " )" );
end );
#############################################################################
##
#M ViewObj( <smg cong> )
##
## left semigroup congruence
InstallMethod( ViewObj,
"for a LeftSemigroupCongruence",
true,
[ IsLeftSemigroupCongruence ], 0,
function( S )
Print( "<LeftSemigroupCongruence>" );
end );
InstallMethod( ViewObj,
"for a LeftSemigroupCongruence with known generating pairs",
true,
[ IsLeftSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "<LeftSemigroupCongruence with ",
Length( GeneratingPairsOfMagmaCongruence( S ) ),
" generating pairs>" );
end );
## right semigroup congruence
InstallMethod( ViewObj,
"for a RightSemigrouCongruence",
true,
[ IsRightSemigroupCongruence ], 0,
function( S )
Print( "<RightSemigroupCongruence>" );
end );
InstallMethod( ViewObj,
"for a RightSemigroupCongruence with generators",
true,
[ IsRightSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "<RightSemigroupCongruence with ",
Length( GeneratingPairsOfMagmaCongruence( S ) ),
" generating pairs>" );
end );
## two sided semigroup congruence
InstallMethod( ViewObj,
"for a semigroup congruence",
true,
[ IsSemigroupCongruence ], 0,
function( S )
Print( "<semigroup congruence>" );
end );
InstallMethod( ViewObj,
"for a semigroup Congruence with known generating pairs",
true,
[ IsSemigroupCongruence and HasGeneratingPairsOfMagmaCongruence ], 0,
function( S )
Print( "<semigroup congruence with ",
Length(GeneratingPairsOfMagmaCongruence( S )),
" generating pairs>" );
end );
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