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Merge pull request #325 from wilfwilson/rzms-idempotent-subsemigroup
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Install specific IsIdempotentGenerated methods for RMS/RZMS
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@mtorpey
mtorpey authored Jun 14, 2017
2 parents 734f649 + 4d17f82 commit 634277b
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Showing 5 changed files with 356 additions and 10 deletions.
73 changes: 73 additions & 0 deletions gap/attributes/attr.gi
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Expand Up @@ -555,6 +555,79 @@ function(S)
return out;
end);

InstallMethod(IdempotentGeneratedSubsemigroup,
"for a Rees matrix subsemigroup",
[IsReesMatrixSubsemigroup],
function(R)
local mat, I, J, nrrows, nrcols, min, max, gens, out, i;

if not IsFinite(R) or not IsReesMatrixSemigroup(R)
or not IsGroup(UnderlyingSemigroup(R)) then
TryNextMethod();
fi;

mat := Matrix(R);
I := Rows(R);
J := Columns(R);
nrrows := Length(I);
nrcols := Length(J);

min := Minimum(nrrows, nrcols);
max := Maximum(nrrows, nrcols);
gens := EmptyPlist(max);
for i in [1 .. min] do
Add(gens, RMSElement(R, I[i], mat[J[i]][I[i]] ^ -1, J[i]));
od;
for i in [min + 1 .. nrrows] do
Add(gens, RMSElement(R, I[i], mat[J[1]][I[i]] ^ -1, J[1]));
od;
for i in [min + 1 .. nrcols] do
Add(gens, RMSElement(R, I[1], mat[J[i]][I[1]] ^ -1, J[i]));
od;
out := Semigroup(gens);
SetIsRegularSemigroup(out, true);
SetIsIdempotentGenerated(out, true);
SetIsSimpleSemigroup(out, true);
return out;
end);

InstallMethod(IdempotentGeneratedSubsemigroup,
"for a Rees 0-matrix subsemigroup",
[IsReesZeroMatrixSubsemigroup],
function(R)
local mat, gr, gens, I, J, nrrows, k, out, i, j;

if not IsFinite(R) or not IsReesZeroMatrixSemigroup(R)
or not IsGroup(UnderlyingSemigroup(R)) then
TryNextMethod();
fi;

mat := Matrix(R);
gr := RZMSDigraph(R);
gens := [];

if IsCompleteBipartiteDigraph(ReducedDigraph(gr)) or IsEmptyDigraph(gr) then
Add(gens, MultiplicativeZero(R));
fi;

gr := OutNeighbours(UndirectedSpanningForest(gr));
I := Rows(R);
J := Columns(R);
nrrows := Length(I);

for i in [1 .. nrrows] do
for j in gr[i] do
k := J[j - nrrows];
Add(gens, RMSElement(R, I[i], mat[k][I[i]] ^ -1, k));
od;
od;

out := Semigroup(gens);
SetIsRegularSemigroup(out, true);
SetIsIdempotentGenerated(out, true);
return out;
end);

InstallMethod(InjectionPrincipalFactor, "for a Green's D-class (Semigroups)",
[IsGreensDClass],
function(D)
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9 changes: 6 additions & 3 deletions gap/attributes/maximal.gi
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Expand Up @@ -1061,10 +1061,13 @@ function(R, opts)
if not failed and not opts.number then
Info(InfoSemigroups, 2, "...creating these maximal subsemigroups.");

# TODO only need generating set for idempotents - don't not *every* idem
idems := ShallowCopy(Idempotents(R));
idems := GeneratorsOfSemigroup(IdempotentGeneratedSubsemigroup(R));
idems := ShallowCopy(idems);
if not opts.zero or not IsCompleteBipartiteDigraph(RZMSDigraph(R_n)) then
Remove(idems, Position(idems, zero));
x := Position(idems, zero);
if x <> fail then
Remove(idems, x);
fi;
fi;

# Max subsemigroup arising from <max[i]> <--> Transversal of <results[i]>
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2 changes: 1 addition & 1 deletion tst/extreme/maximal.tst
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Expand Up @@ -281,7 +281,7 @@ gap> max := MaximalSubsemigroups(R1);
<subsemigroup of 5x5 Rees 0-matrix semigroup with 10 generators>,
<subsemigroup of 5x5 Rees 0-matrix semigroup with 10 generators>,
<subsemigroup of 5x5 Rees 0-matrix semigroup with 10 generators>,
<subsemigroup of 5x5 Rees 0-matrix semigroup with 11 generators> ]
<subsemigroup of 5x5 Rees 0-matrix semigroup with 10 generators> ]
gap> NrMaximalSubsemigroups(R1);
26

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262 changes: 262 additions & 0 deletions tst/standard/attr.tst
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Expand Up @@ -1400,6 +1400,268 @@ Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 2nd choice method found for `InversesOfSemigroupElementNC' on 2 argu\
ments

#T# attr: IdempotentGeneratedSubsemigroup, 2
gap> S := FullTransformationMonoid(3);;
gap> I := IdempotentGeneratedSubsemigroup(S);;
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true

#T# attr: IdempotentGeneratedSubsemigroup, for an Rees matrix semigroup, 1
# TryNextMethod()

# Error: infinite
gap> S := FreeSemigroup(1);
<free semigroup on the generators [ s1 ]>
gap> R := ReesMatrixSemigroup(S, [[S.1]]);
<Rees matrix semigroup 1x1 over <free semigroup on the generators [ s1 ]>>
gap> IdempotentGeneratedSubsemigroup(R);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 4th choice method found for `IdempotentGeneratedSubsemigroup' on 1 a\
rguments
# not a Rees matrix semigroup
gap> x := Transformation([2, 2]);;
gap> R := ReesMatrixSemigroup(FullTransformationMonoid(2),
> [[IdentityTransformation, x],
> [x, x]]);
<Rees matrix semigroup 2x2 over <full transformation monoid of degree 2>>
gap> S := Semigroup(RMSElement(R, 1, IdentityTransformation, 1),
> RMSElement(R, 2, IdentityTransformation, 2));
<subsemigroup of 2x2 Rees matrix semigroup with 2 generators>
gap> IsReesMatrixSubsemigroup(S);
true
gap> IsReesMatrixSemigroup(S);
false
gap> I := IdempotentGeneratedSubsemigroup(S);;
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> Size(I) = 5;
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
# not over a group
gap> x := Transformation([2, 2]);;
gap> R := ReesMatrixSemigroup(FullTransformationMonoid(2),
> [[IdentityTransformation, x],
> [x, x]]);
<Rees matrix semigroup 2x2 over <full transformation monoid of degree 2>>
gap> I := IdempotentGeneratedSubsemigroup(R);;
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> Size(I) = 9;
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
#T# attr: IdempotentGeneratedSubsemigroup, for an Rees matrix semigroup, 2
# Rectangular bands
gap> R := RectangularBand(IsReesMatrixSemigroup, 1, 1);
<Rees matrix semigroup 1x1 over Group(())>
gap> I := IdempotentGeneratedSubsemigroup(R);
<subsemigroup of 1x1 Rees matrix semigroup with 1 generator>
gap> I = R;
true
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> R := RectangularBand(IsReesMatrixSemigroup, 3, 1);
<Rees matrix semigroup 3x1 over Group(())>
gap> I := IdempotentGeneratedSubsemigroup(R);
<subsemigroup of 3x1 Rees matrix semigroup with 3 generators>
gap> I = R;
true
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> R := RectangularBand(IsReesMatrixSemigroup, 1, 3);
<Rees matrix semigroup 1x3 over Group(())>
gap> I := IdempotentGeneratedSubsemigroup(R);
<subsemigroup of 1x3 Rees matrix semigroup with 3 generators>
gap> I = R;
true
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> R := RectangularBand(IsReesMatrixSemigroup, 2, 2);
<Rees matrix semigroup 2x2 over Group(())>
gap> I := IdempotentGeneratedSubsemigroup(R);
<subsemigroup of 2x2 Rees matrix semigroup with 2 generators>
gap> I = R;
true
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
# Subsemigroup, giving non-standard matrix
gap> x := [[(2, 3, 5), (1, 2, 4)(3, 5), (1, 3, 5, 2, 4)],
> [(1, 3, 5, 4), (2, 3, 5, 4), (1, 3, 5)(2, 4)],
> [(2, 3, 5), (1, 5, 4, 3, 2), ()],
> [(1, 4, 2, 5), (1, 2)(3, 5), (1, 5, 3, 2, 4)],
> [(1, 4)(2, 3, 5), (1, 2)(4, 5), (1, 3, 4, 2, 5)]];;
gap> G := SymmetricGroup(5);;
gap> R := ReesMatrixSemigroup(G, x);
<Rees matrix semigroup 3x5 over Sym( [ 1 .. 5 ] )>
gap> S := ReesMatrixSubsemigroup(R, [2], G, [4]);
<Rees matrix semigroup 1x1 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees matrix semigroup with 1 generator>
gap> GeneratorsOfSemigroup(I);
[ (2,(1,2)(3,5),4) ]
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> S := ReesMatrixSubsemigroup(R, [2], G, [3, 4, 5]);
<Rees matrix semigroup 1x3 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees matrix semigroup with 3 generators>
gap> GeneratorsOfSemigroup(I);
[ (2,(1,2,3,4,5),3), (2,(1,2)(3,5),4), (2,(1,2)(4,5),5) ]
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> S := ReesMatrixSubsemigroup(R, [2, 3], G, [4]);
<Rees matrix semigroup 2x1 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees matrix semigroup with 2 generators>
gap> GeneratorsOfSemigroup(I);
[ (2,(1,2)(3,5),4), (3,(1,4,2,3,5),4) ]
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> S := ReesMatrixSubsemigroup(R, [2, 3], G, [4, 5]);
<Rees matrix semigroup 2x2 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees matrix semigroup with 2 generators>
gap> GeneratorsOfSemigroup(I);
[ (2,(1,2)(3,5),4), (3,(1,5,2,4,3),5) ]
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
#T# attr: IdempotentGeneratedSubsemigroup, for an Rees 0-matrix semigroup, 1
# TryNextMethod()
# Error: infinite
gap> S := FreeSemigroup(1);
<free semigroup on the generators [ s1 ]>
gap> R := ReesZeroMatrixSemigroup(S, [[S.1]]);
<Rees 0-matrix semigroup 1x1 over <free semigroup on the generators [ s1 ]>>
gap> IdempotentGeneratedSubsemigroup(R);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 4th choice method found for `IdempotentGeneratedSubsemigroup' on 1 a\
rguments

# not a Rees matrix semigroup
gap> x := Transformation([2, 2]);;
gap> R := ReesZeroMatrixSemigroup(FullTransformationMonoid(2),
> [[IdentityTransformation, 0],
> [0, x]]);
<Rees 0-matrix semigroup 2x2 over <full transformation monoid of degree 2>>
gap> S := Semigroup(RMSElement(R, 1, IdentityTransformation, 1),
> RMSElement(R, 2, IdentityTransformation, 2));
<subsemigroup of 2x2 Rees 0-matrix semigroup with 2 generators>
gap> IsReesZeroMatrixSubsemigroup(S);
true
gap> IsReesZeroMatrixSemigroup(S);
false
gap> I := IdempotentGeneratedSubsemigroup(S);;
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> Size(I) = 3;
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true

# not over a group
gap> x := Transformation([2, 2]);;
gap> R := ReesZeroMatrixSemigroup(FullTransformationMonoid(2),
> [[IdentityTransformation, 0],
> [x, x]]);
<Rees 0-matrix semigroup 2x2 over <full transformation monoid of degree 2>>
gap> I := IdempotentGeneratedSubsemigroup(R);;
gap> HasIsIdempotentGenerated(I);
true
gap> IsIdempotentGenerated(I);
true
gap> Size(I) = 10;
true
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true

#T# attr: IdempotentGeneratedSubsemigroup, for an Rees 0-matrix semigroup, 2

# Subsemigroup, giving non-standard matrix, 1
gap> R := ReesZeroMatrixSemigroup(Group(()), [[(), ()], [(), ()]]);
<Rees 0-matrix semigroup 2x2 over Group(())>
gap> S := Semigroup(RMSElement(R, 2, (), 2), MultiplicativeZero(R));
<subsemigroup of 2x2 Rees 0-matrix semigroup with 2 generators>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 2x2 Rees 0-matrix semigroup with 2 generators>
gap> I = S;
true
gap> Elements(I);
[ 0, (2,(),2) ]

# Subsemigroup, giving non-standard matrix, 2
gap> x := [[(1, 3, 5)(2, 4), (1, 4, 3, 2, 5), (1, 3, 5)(2, 4)],
> [(2, 4, 5, 3), 0, (1, 4, 5, 3)],
> [(3, 5, 4), 0, (1, 4, 5)(2, 3)],
> [0, 0, (1, 3, 4, 2)],
> [(2, 3, 4), (1, 2, 5, 4, 3), (1, 5, 3)(2, 4)]];;
gap> G := SymmetricGroup(5);;
gap> R := ReesZeroMatrixSemigroup(G, x);
<Rees 0-matrix semigroup 3x5 over Sym( [ 1 .. 5 ] )>
gap> S := ReesZeroMatrixSubsemigroup(R, [2], G, [4]);
<Rees 0-matrix semigroup 1x1 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees 0-matrix semigroup with 1 generator>
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> I = Semigroup(Idempotents(S));
true
gap> S := ReesZeroMatrixSubsemigroup(R, [2], G, [3, 4, 5]);
<Rees 0-matrix semigroup 1x3 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees 0-matrix semigroup with 2 generators>
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> I = Semigroup(Idempotents(S));
true
gap> S := ReesZeroMatrixSubsemigroup(R, [2, 3], G, [4]);
<Rees 0-matrix semigroup 2x1 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees 0-matrix semigroup with 2 generators>
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> I = Semigroup(Idempotents(S));
true
gap> S := ReesZeroMatrixSubsemigroup(R, [2, 3], G, [4, 5]);
<Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 5 ] )>
gap> I := IdempotentGeneratedSubsemigroup(S);
<subsemigroup of 3x5 Rees 0-matrix semigroup with 3 generators>
gap> ForAll(GeneratorsOfSemigroup(I), IsIdempotent);
true
gap> I = Semigroup(Idempotents(S));
true

#T# SEMIGROUPS_UnbindVariables
gap> Unbind(D);
gap> Unbind(G);
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