#############################################################################
##
#W extreme/ideals.tst
#Y Copyright (C) 2013-15 James D. Mitchell
## Julius Jonusas
## Wilf A. Wilson
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
#@local A, B, C, I, J, K, L, M, R, S, T, U, a, acting, an, data, gens, gensI
#@local gensJ, gensK, gensL, ideals, idems, o, s, x
gap> START_TEST("Semigroups package: extreme/ideals.tst");
gap> LoadPackage("semigroups", false);;
#
gap> SEMIGROUPS.StartTest();
gap> SEMIGROUPS.DefaultOptionsRec.acting := true;;
# IdealsTest1
gap> gens := [Transformation([2, 6, 1, 7, 5, 3, 4]),
> Transformation([5, 3, 7, 2, 1, 6, 4]),
> Transformation([2, 5, 5, 3, 4, 2, 3]),
> Transformation([1, 5, 1, 6, 1, 5, 6]),
> Transformation([6, 2, 2, 2, 5, 1, 2]),
> Transformation([7, 5, 4, 4, 4, 5, 5]),
> Transformation([5, 1, 6, 1, 1, 5, 1]),
> Transformation([3, 5, 2, 3, 2, 2, 3])];;
gap> S := Semigroup(gens);;
gap> I := SemigroupIdeal(S, S.8);;
gap> x := S.8 * S.1 * S.4 * S.3;;
gap> R := RClass(I, x);;
gap> x in R;
true
gap> I := MinimalIdeal(I);
<simple transformation semigroup ideal of degree 7 with 1 generator>
gap> IsRegularSemigroup(I);
true
gap> Idempotents(I, 0);
[ ]
gap> idems := ShallowCopy(Idempotents(I));;
gap> Sort(idems);
gap> idems;
[ Transformation( [ 1, 1, 1, 1, 1, 1, 1 ] ),
Transformation( [ 2, 2, 2, 2, 2, 2, 2 ] ),
Transformation( [ 3, 3, 3, 3, 3, 3, 3 ] ),
Transformation( [ 4, 4, 4, 4, 4, 4, 4 ] ),
Transformation( [ 5, 5, 5, 5, 5, 5, 5 ] ),
Transformation( [ 6, 6, 6, 6, 6, 6, 6 ] ),
Transformation( [ 7, 7, 7, 7, 7, 7, 7 ] ) ]
gap> Idempotents(I, 2);
[ ]
gap> x in R;
true
gap> I := MinimalIdeal(I);
<simple transformation semigroup ideal of degree 7 with 1 generator>
gap> IsRegularSemigroup(I);;
gap> Idempotents(I, 0);
[ ]
gap> idems := ShallowCopy(Idempotents(I));;
gap> Sort(idems);
gap> idems;
[ Transformation( [ 1, 1, 1, 1, 1, 1, 1 ] ),
Transformation( [ 2, 2, 2, 2, 2, 2, 2 ] ),
Transformation( [ 3, 3, 3, 3, 3, 3, 3 ] ),
Transformation( [ 4, 4, 4, 4, 4, 4, 4 ] ),
Transformation( [ 5, 5, 5, 5, 5, 5, 5 ] ),
Transformation( [ 6, 6, 6, 6, 6, 6, 6 ] ),
Transformation( [ 7, 7, 7, 7, 7, 7, 7 ] ) ]
gap> Idempotents(I, 2);
[ ]
gap> S := Semigroup([Transformation([1, 3, 4, 1, 3, 5]),
> Transformation([1, 5, 3, 5, 5, 5]),
> Transformation([2, 4, 6, 1, 6, 5]),
> Transformation([3, 2, 4, 2, 3, 3]),
> Transformation([4, 1, 2, 6, 2, 1]),
> Transformation([4, 6, 4, 3, 3, 3]),
> Transformation([4, 6, 5, 5, 2, 6]),
> Transformation([5, 1, 6, 1, 6, 3]),
> Transformation([5, 2, 5, 3, 5, 3]),
> Transformation([6, 4, 5, 5, 1, 6])]);
<transformation semigroup of degree 6 with 10 generators>
gap> I := SemigroupIdeal(S, Representative(DClasses(S)[3]));
<non-regular transformation semigroup ideal of degree 6 with 1 generator>
gap> S := JonesMonoid(6);
<regular bipartition *-monoid of degree 6 with 5 generators>
gap> a := Bipartition([[1, -1], [2, 3], [4, -2], [5, -5], [6, -6],
> [-3, -4]]);;
gap> I := SemigroupIdeal(S, a);
<regular bipartition *-semigroup ideal of degree 6 with 1 generator>
gap> a := Bipartition([[1, 4], [2, 3], [5, 6], [-1, -2],
> [-3, -6], [-4, -5]]);;
gap> InversesOfSemigroupElement(I, a);
[ <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ -1, -2 ], [ -3, -4 ],
[ -5, -6 ]>, <bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, 6 ], [ -1, -2 ],
[ -3, -4 ], [ -5, -6 ]>,
<bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -2 ], [ -3, -4 ],
[ -5, -6 ]>, <bipartition: [ 1, 6 ], [ 2, 3 ], [ 4, 5 ], [ -1, -2 ],
[ -3, -4 ], [ -5, -6 ]>,
<bipartition: [ 1, 6 ], [ 2, 5 ], [ 3, 4 ], [ -1, -2 ], [ -3, -4 ],
[ -5, -6 ]>, <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ -1, -4 ],
[ -2, -3 ], [ -5, -6 ]>,
<bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, 6 ], [ -1, -4 ], [ -2, -3 ],
[ -5, -6 ]>, <bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -4 ],
[ -2, -3 ], [ -5, -6 ]>,
<bipartition: [ 1, 6 ], [ 2, 3 ], [ 4, 5 ], [ -1, -4 ], [ -2, -3 ],
[ -5, -6 ]>, <bipartition: [ 1, 6 ], [ 2, 5 ], [ 3, 4 ], [ -1, -4 ],
[ -2, -3 ], [ -5, -6 ]>,
<bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ -1, -2 ], [ -3, -6 ],
[ -4, -5 ]>, <bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, 6 ], [ -1, -2 ],
[ -3, -6 ], [ -4, -5 ]>,
<bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -2 ], [ -3, -6 ],
[ -4, -5 ]>, <bipartition: [ 1, 6 ], [ 2, 3 ], [ 4, 5 ], [ -1, -2 ],
[ -3, -6 ], [ -4, -5 ]>,
<bipartition: [ 1, 6 ], [ 2, 5 ], [ 3, 4 ], [ -1, -2 ], [ -3, -6 ],
[ -4, -5 ]>, <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ -1, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, 6 ], [ -1, -6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, 6 ], [ 2, 3 ], [ 4, 5 ], [ -1, -6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, 6 ], [ 2, 5 ], [ 3, 4 ], [ -1, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, 6 ], [ -1, -6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, 6 ], [ -1, -6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, 2 ], [ 3, 6 ], [ 4, 5 ], [ -1, -6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, 6 ], [ 2, 3 ], [ 4, 5 ], [ -1, -6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, 6 ], [ 2, 5 ], [ 3, 4 ], [ -1, -6 ], [ -2, -5 ],
[ -3, -4 ]> ]
gap> a := Bipartition([[1, 2], [3, 4], [5, -1], [6, -6], [-2, -3],
> [-4, -5]]);;
gap> InversesOfSemigroupElement(I, a);
[ <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, -5 ], [ 6, -6 ], [ -1, -2 ],
[ -3, -4 ]>, <bipartition: [ 1, 2 ], [ 3, -5 ], [ 4, 5 ], [ 6, -6 ],
[ -1, -2 ], [ -3, -4 ]>,
<bipartition: [ 1, 2 ], [ 3, -5 ], [ 4, -6 ], [ 5, 6 ], [ -1, -2 ],
[ -3, -4 ]>, <bipartition: [ 1, -5 ], [ 2, 3 ], [ 4, 5 ], [ 6, -6 ],
[ -1, -2 ], [ -3, -4 ]>,
<bipartition: [ 1, -5 ], [ 2, 3 ], [ 4, -6 ], [ 5, 6 ], [ -1, -2 ],
[ -3, -4 ]>, <bipartition: [ 1, -5 ], [ 2, -6 ], [ 3, 4 ], [ 5, 6 ],
[ -1, -2 ], [ -3, -4 ]>,
<bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, -5 ], [ 6, -6 ], [ -1, -2 ],
[ -3, -4 ]>, <bipartition: [ 1, -5 ], [ 2, 5 ], [ 3, 4 ], [ 6, -6 ],
[ -1, -2 ], [ -3, -4 ]>,
<bipartition: [ 1, -5 ], [ 2, -6 ], [ 3, 6 ], [ 4, 5 ], [ -1, -2 ],
[ -3, -4 ]>, <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, -3 ], [ 6, -6 ],
[ -1, -2 ], [ -4, -5 ]>,
<bipartition: [ 1, 2 ], [ 3, -3 ], [ 4, 5 ], [ 6, -6 ], [ -1, -2 ],
[ -4, -5 ]>, <bipartition: [ 1, 2 ], [ 3, -3 ], [ 4, -6 ], [ 5, 6 ],
[ -1, -2 ], [ -4, -5 ]>,
<bipartition: [ 1, -3 ], [ 2, 3 ], [ 4, 5 ], [ 6, -6 ], [ -1, -2 ],
[ -4, -5 ]>, <bipartition: [ 1, -3 ], [ 2, 3 ], [ 4, -6 ], [ 5, 6 ],
[ -1, -2 ], [ -4, -5 ]>,
<bipartition: [ 1, -3 ], [ 2, -6 ], [ 3, 4 ], [ 5, 6 ], [ -1, -2 ],
[ -4, -5 ]>, <bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, -3 ], [ 6, -6 ],
[ -1, -2 ], [ -4, -5 ]>,
<bipartition: [ 1, -3 ], [ 2, 5 ], [ 3, 4 ], [ 6, -6 ], [ -1, -2 ],
[ -4, -5 ]>, <bipartition: [ 1, -3 ], [ 2, -6 ], [ 3, 6 ], [ 4, 5 ],
[ -1, -2 ], [ -4, -5 ]>,
<bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, -1 ], [ 6, -6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, 2 ], [ 3, -1 ], [ 4, 5 ], [ 6, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, 2 ], [ 3, -1 ], [ 4, -6 ], [ 5, 6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, -1 ], [ 2, 3 ], [ 4, 5 ], [ 6, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, -1 ], [ 2, 3 ], [ 4, -6 ], [ 5, 6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, -1 ], [ 2, -6 ], [ 3, 4 ], [ 5, 6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, -1 ], [ 6, -6 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, -1 ], [ 2, 5 ], [ 3, 4 ], [ 6, -6 ],
[ -2, -3 ], [ -4, -5 ]>,
<bipartition: [ 1, -1 ], [ 2, -6 ], [ 3, 6 ], [ 4, 5 ], [ -2, -3 ],
[ -4, -5 ]>, <bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, -5 ], [ 6, -6 ],
[ -1, -4 ], [ -2, -3 ]>,
<bipartition: [ 1, 2 ], [ 3, -5 ], [ 4, 5 ], [ 6, -6 ], [ -1, -4 ],
[ -2, -3 ]>, <bipartition: [ 1, 2 ], [ 3, -5 ], [ 4, -6 ], [ 5, 6 ],
[ -1, -4 ], [ -2, -3 ]>,
<bipartition: [ 1, -5 ], [ 2, 3 ], [ 4, 5 ], [ 6, -6 ], [ -1, -4 ],
[ -2, -3 ]>, <bipartition: [ 1, -5 ], [ 2, 3 ], [ 4, -6 ], [ 5, 6 ],
[ -1, -4 ], [ -2, -3 ]>,
<bipartition: [ 1, -5 ], [ 2, -6 ], [ 3, 4 ], [ 5, 6 ], [ -1, -4 ],
[ -2, -3 ]>, <bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, -5 ], [ 6, -6 ],
[ -1, -4 ], [ -2, -3 ]>,
<bipartition: [ 1, -5 ], [ 2, 5 ], [ 3, 4 ], [ 6, -6 ], [ -1, -4 ],
[ -2, -3 ]>, <bipartition: [ 1, -5 ], [ 2, -6 ], [ 3, 6 ], [ 4, 5 ],
[ -1, -4 ], [ -2, -3 ]>,
<bipartition: [ 1, 2 ], [ 3, 4 ], [ 5, -1 ], [ 6, -6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, 2 ], [ 3, -1 ], [ 4, 5 ], [ 6, -6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, 2 ], [ 3, -1 ], [ 4, -6 ], [ 5, 6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, -1 ], [ 2, 3 ], [ 4, 5 ], [ 6, -6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, -1 ], [ 2, 3 ], [ 4, -6 ], [ 5, 6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, -1 ], [ 2, -6 ], [ 3, 4 ], [ 5, 6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, 4 ], [ 2, 3 ], [ 5, -1 ], [ 6, -6 ], [ -2, -5 ],
[ -3, -4 ]>, <bipartition: [ 1, -1 ], [ 2, 5 ], [ 3, 4 ], [ 6, -6 ],
[ -2, -5 ], [ -3, -4 ]>,
<bipartition: [ 1, -1 ], [ 2, -6 ], [ 3, 6 ], [ 4, 5 ], [ -2, -5 ],
[ -3, -4 ]> ]
gap> S := SymmetricInverseMonoid(8);;
gap> x := PartialPerm([1]);;
gap> I := SemigroupIdeal(S, x);
<inverse partial perm semigroup ideal of rank 8 with 1 generator>
gap> IsZeroSimpleSemigroup(I);
true
# IdealsTest2
gap> gens := [Transformation([3, 4, 1, 2, 1]),
> Transformation([4, 2, 1, 5, 5]),
> Transformation([4, 2, 2, 2, 4])];;
gap> s := Semigroup(gens);;
gap> I := SemigroupIdeal(s, gens);
<regular transformation semigroup ideal of degree 5 with 3 generators>
gap> data := SemigroupData(I);
<closed semigroup ideal data with 26 reps, 23 lambda-values, 26 rho-values>
gap> Size(I);
731
gap> NrDClasses(I);
4
gap> GreensDClasses(I);
[ <Green's D-class: Transformation( [ 3, 4, 1, 2, 1 ] )>,
<Green's D-class: Transformation( [ 4, 2, 2, 2, 4 ] )>,
<Green's D-class: Transformation( [ 4, 5, 2, 4, 4 ] )>,
<Green's D-class: Transformation( [ 2, 2, 2, 2, 2 ] )> ]
# IdealsTest3
gap> gens := [Transformation([1, 3, 4, 1]),
> Transformation([2, 4, 1, 2]),
> Transformation([3, 1, 1, 3]),
> Transformation([3, 3, 4, 1])];;
gap> s := Semigroup(gens);;
gap> I := SemigroupIdeal(s, gens{[1, 2]});
<non-regular transformation semigroup ideal of degree 4 with 2 generators>
gap> o := LambdaOrb(I);
<closed ideal lambda orbit with 11 points in 2 components>
# IdealsTest4
gap> gens := [Transformation([1, 3, 2, 3]),
> Transformation([1, 4, 1, 2]),
> Transformation([2, 4, 1, 1]),
> Transformation([3, 4, 2, 2])];;
gap> s := Semigroup(gens);;
gap> I := SemigroupIdeal(s, [gens[2] * gens[1], gens[3] ^ 3]);
<non-regular transformation semigroup ideal of degree 4 with 2 generators>
gap> o := RhoOrb(I);
<closed ideal rho orbit with 10 points in 2 components>
# IdealsTest5
gap> gens := [
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 6, 7, 3, 8, 2, 9, 5]),
> PartialPerm([1, 2, 7, 9], [5, 6, 4, 3]),
> PartialPerm([1, 2, 6, 7, 8], [5, 1, 6, 2, 3])];;
gap> s := Semigroup(gens);
<partial perm semigroup of rank 10 with 3 generators>
gap> I := SemigroupIdeal(s, [gens[1] ^ 2, gens[2]]);
<non-regular partial perm semigroup ideal of rank 10 with 2 generators>
gap> R := GreensRClassOfElement(I, gens[1] ^ 2);
<Green's R-class: [1,3,9][4,7][10,8](2)(6)>
gap> DClassOfRClass(R);
<Green's D-class: [1,3,9][4,7][10,8](2)(6)>
gap> L := GreensLClassOfElement(I, gens[1] ^ 2);
<Green's L-class: [1,3,9][4,7][10,8](2)(6)>
gap> DClassOfLClass(L);
<Green's D-class: [1,3,9][4,7][10,8](2)(6)>
# IdealsTest6: \in for an inverse op semigroup ideal
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 5, 6, 7, 8], [5, 9, 10, 6, 3, 8, 4]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8], [6, 2, 8, 4, 7, 5, 3]),
> PartialPerm([1, 2, 4, 6, 8, 9], [7, 10, 1, 9, 4, 2]),
> PartialPerm([1, 2, 4, 7, 8, 9], [10, 7, 8, 5, 9, 1]),
> PartialPerm([1, 2, 3, 6, 8, 10], [2, 6, 7, 9, 1, 5]),
> PartialPerm([1, 2, 3, 4, 5, 8, 10], [3, 1, 4, 2, 5, 6, 7]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 10], [3, 8, 1, 9, 4, 10, 5, 6]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 7, 6, 9, 10, 1, 3, 2]),
> PartialPerm([1, 2, 4, 5, 8, 10], [6, 2, 7, 8, 10, 4]),
> PartialPerm([1, 2, 3, 4, 5, 8, 10], [7, 1, 4, 3, 2, 6, 5]));;
gap> I := SemigroupIdeal(S,
> [PartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 7, 6, 9, 10, 1, 3, 2]),
> PartialPerm([4], [3]), PartialPerm([6, 7], [5, 8])]);;
gap> Size(I);
4626941
# IdealsTest7: attributes.gi
gap> S := Semigroup([Transformation([1, 3, 4, 1, 3, 5]),
> Transformation([1, 5, 3, 5, 5, 5]),
> Transformation([2, 4, 6, 1, 6, 5]),
> Transformation([3, 2, 4, 2, 3, 3]),
> Transformation([4, 1, 2, 6, 2, 1]),
> Transformation([4, 6, 4, 3, 3, 3]),
> Transformation([4, 6, 5, 5, 2, 6]),
> Transformation([5, 1, 6, 1, 6, 3]),
> Transformation([5, 2, 5, 3, 5, 3]),
> Transformation([6, 4, 5, 5, 1, 6])]);;
gap> I := SemigroupIdeal(S, Transformation([1, 3, 4, 1, 3, 5]));;
gap> J := SemigroupIdeal(S, Transformation([1, 5, 3, 5, 5, 5]));;
gap> IsGreensDGreaterThanFunc(I); IsGreensDGreaterThanFunc(J);
function( x, y ) ... end
function( x, y ) ... end
gap> Length(MaximalDClasses(I));
265
gap> MaximalDClasses(J);
[ <Green's D-class: Transformation( [ 1, 5, 3, 5, 5, 5 ] )> ]
gap> StructureDescriptionSchutzenbergerGroups(I);
[ "1", "C2", "S3" ]
gap> StructureDescriptionSchutzenbergerGroups(J);
[ "1", "C2", "S3" ]
gap> StructureDescriptionMaximalSubgroups(J);
[ "1", "C2", "S3" ]
gap> StructureDescriptionMaximalSubgroups(I);
[ "1", "C2", "S3" ]
gap> GroupOfUnits(I);
fail
gap> GroupOfUnits(J);
fail
gap> IdempotentGeneratedSubsemigroup(I);;
gap> IdempotentGeneratedSubsemigroup(J);;
gap> last = last2;
true
gap> x := Transformation([5, 5, 1, 4, 1, 1]);;
gap> x in I;
true
gap> x in J;
true
gap> Length(InversesOfSemigroupElement(I, x)) = 84;
true
gap> Set(InversesOfSemigroupElement(J, x)) =
> Set(InversesOfSemigroupElement(I, x));
true
gap> MultiplicativeNeutralElement(I);
fail
gap> MultiplicativeNeutralElement(J);
fail
gap> MultiplicativeNeutralElement(I);
fail
gap> MultiplicativeNeutralElement(J);
fail
gap> MultiplicativeZero(I);
fail
gap> MultiplicativeZero(J);
fail
gap> MinimalIdeal(I);
<simple transformation semigroup ideal of degree 6 with 1 generator>
gap> MinimalIdeal(J);
<simple transformation semigroup ideal of degree 6 with 1 generator>
gap> last = last2;
true
gap> MinimalDClass(I);
<Green's D-class: Transformation( [ 5, 5, 5, 5, 5, 5 ] )>
gap> MinimalDClass(J);
<Green's D-class: Transformation( [ 5, 5, 5, 5, 5, 5 ] )>
# IdealsTest8: attributes
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 5, 6, 7, 8], [5, 9, 10, 6, 3, 8, 4]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8], [6, 2, 8, 4, 7, 5, 3]),
> PartialPerm([1, 2, 4, 6, 8, 9], [7, 10, 1, 9, 4, 2]),
> PartialPerm([1, 2, 4, 7, 8, 9], [10, 7, 8, 5, 9, 1]),
> PartialPerm([1, 2, 3, 6, 8, 10], [2, 6, 7, 9, 1, 5]),
> PartialPerm([1, 2, 3, 4, 5, 8, 10], [3, 1, 4, 2, 5, 6, 7]),
> PartialPerm([1, 2, 3, 4, 6, 7, 8, 10], [3, 8, 1, 9, 4, 10, 5, 6]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 10], [4, 7, 6, 9, 10, 1, 3, 2]),
> PartialPerm([1, 2, 4, 5, 8, 10], [6, 2, 7, 8, 10, 4]),
> PartialPerm([1, 2, 3, 4, 5, 8, 10], [7, 1, 4, 3, 2, 6, 5]));;
gap> K := SemigroupIdeal(S, S.1 * S.2 ^ 2 * S.8);
<inverse partial perm semigroup ideal of rank 10 with 1 generator>
gap> MaximalDClasses(K);
[ <Green's D-class: <identity partial perm on [ 3, 6, 9 ]>> ]
gap> StructureDescriptionMaximalSubgroups(K);
[ "1", "C2", "S3" ]
gap> StructureDescriptionSchutzenbergerGroups(K);
[ "1", "C2", "S3" ]
gap> GroupOfUnits(K);
fail
gap> Size(IdempotentGeneratedSubsemigroup(K));
176
gap> x := PartialPerm([2, 4, 6, 8, 9], [10, 1, 9, 4, 2]);;
gap> x in K;
false
gap> x := PartialPerm([9], [9]);;
gap> x in K;
true
gap> InversesOfSemigroupElement(K, x);
[ <identity partial perm on [ 9 ]> ]
gap> MultiplicativeZero(K);
<empty partial perm>
gap> MultiplicativeNeutralElement(K);
fail
gap> MinimalIdeal(K);
<partial perm group of rank 0>
gap> MinimalDClass(K);
<Green's D-class: <empty partial perm>>
gap> I := MinimalIdeal(K);
<partial perm group of rank 0>
gap> x := IsomorphismPermGroup(I);;
gap> IsTrivial(Range(x));
true
# IdealsTest9: attributes.gi
gap> S := Monoid(Transformation([2, 6, 7, 2, 6, 1, 1, 5]),
> Transformation([3, 8, 1, 4, 5, 6, 7, 1]),
> Transformation([4, 3, 2, 7, 7, 6, 6, 5]),
> Transformation([7, 1, 7, 4, 2, 5, 6, 3]));;
gap> L := SemigroupIdeal(S, GeneratorsOfSemigroup(S));
<non-regular transformation semigroup ideal of degree 8 with 5 generators>
gap> Length(MaximalDClasses(L));
1
gap> L = S;
true
gap> MaximalDClasses(L);
[ <Green's D-class: IdentityTransformation> ]
gap> StructureDescriptionSchutzenbergerGroups(L);
[ "1", "C2", "C4", "C5", "S3", "S4" ]
gap> StructureDescriptionMaximalSubgroups(L);
[ "1", "C2", "C4", "C5", "S3", "S4" ]
gap> GroupOfUnits(L);
<trivial transformation group of degree 0 with 1 generator>
gap> IdempotentGeneratedSubsemigroup(L);;
gap> x := Transformation([1, 4, 4, 5, 5, 3, 3, 1]);;
gap> InversesOfSemigroupElement(L, x);
[ ]
gap> InversesOfSemigroupElement(S, x);
[ ]
gap> MultiplicativeNeutralElement(L);
IdentityTransformation
gap> MultiplicativeZero(L);
fail
gap> MinimalIdeal(L);
<simple transformation semigroup ideal of degree 8 with 1 generator>
gap> L := SemigroupIdeal(S, GeneratorsOfSemigroup(S));
<non-regular transformation semigroup ideal of degree 8 with 5 generators>
gap> MinimalIdeal(J);
<simple transformation semigroup ideal of size 6, degree 6 with 1 generator>
gap> MinimalIdeal(L);
<simple transformation semigroup ideal of degree 8 with 1 generator>
gap> MinimalDClass(L);
<Green's D-class: Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1 ] )>
gap> MinimalDClass(S);
<Green's D-class: Transformation( [ 1, 1, 1, 1, 1, 1, 1, 1 ] )>
# IdealsTest10
gap> R := Semigroup([
> Bipartition([[1, 2, 3, 5, -3], [4, 6, 7, -5], [-1], [-2, -4, -7],
> [-6]]),
> Bipartition([[1, 2, 6, -5], [3], [4, 5, -2], [7, -1, -3, -4, -7],
> [-6]]),
> Bipartition([[1, 3, 4, 5, -2, -3, -5], [2, 6, -1, -6, -7], [7],
> [-4]]),
> Bipartition([[1, 3, 4, 6, -7], [2, 5, -1, -5], [7, -2, -3, -4],
> [-6]]),
> Bipartition([[1, 3], [2, 5, 6, -1, -2, -3], [4, 7, -4, -7], [-5],
> [-6]]),
> Bipartition([[1, 4, 5, -1, -2, -4, -6], [2, 3, 7, -3, -5, -7], [6]]),
> Bipartition([[1, -1, -4], [2, 3, 4, 5, 6, 7, -2, -6], [-3, -5, -7]]),
> Bipartition([[1, 7, -6], [2, 3, 4, 5, -1, -2, -4], [6, -3, -5],
> [-7]]),
> Bipartition([[1, 5, -2, -7], [2, 3, 6, -4], [4, -1, -5, -6], [7],
> [-3]]),
> Bipartition([[1, -3, -4], [2], [3, 7, -1, -7], [4, 5, -6],
> [6, -5], [-2]])]);;
gap> gens := [
> Bipartition([[1, 2, 3, 5, -3], [4, 6, 7, -5], [-1],
> [-2, -4, -7], [-6]]),
> Bipartition([[1, 2, 6, -5], [3], [4, 5, -2], [7, -1, -3, -4, -7],
> [-6]]),
> Bipartition([[1, 3], [2, 5, 6, -1, -2, -3], [4, 7, -4, -7], [-5],
> [-6]]),
> Bipartition([[1, -3, -4], [2], [3, 7, -1, -7], [4, 5, -6],
> [6, -5], [-2]])];;
gap> M := SemigroupIdeal(R, gens);
<non-regular bipartition semigroup ideal of degree 7 with 4 generators>
gap> Length(MaximalDClasses(M));
10
gap> Length(GeneratorsOfSemigroup(M)) < 100;
true
gap> StructureDescriptionSchutzenbergerGroups(M);
[ "1" ]
gap> StructureDescriptionMaximalSubgroups(M);
[ "1" ]
gap> Size(IdempotentGeneratedSubsemigroup(M));
1441
gap> GroupOfUnits(M);
fail
gap> x := Bipartition([[1, 2, 3, 4, 5, 6, 7, -3, -5], [-1],
> [-2, -4, -7], [-6]]);;
gap> x in M;
true
gap> Length(InversesOfSemigroupElement(M, x));
875
gap> ForAll(InversesOfSemigroupElement(M, x), y -> y in M);
true
gap> MultiplicativeNeutralElement(M);
fail
gap> MultiplicativeZero(M);
fail
gap> MinimalIdeal(M);
<simple bipartition semigroup ideal of degree 7 with 1 generator>
gap> MinimalDClass(M) = DClass(M, Bipartition([[1, 2, 3, 4, 5, 6, 7], [-1],
> [-2, -4, -7], [-3, -5], [-6]]));
true
gap> MinimalDClass(R);
<Green's D-class: <bipartition: [ 1, 2, 3, 4, 5, 6, 7 ],
[ -1, -2, -3, -4, -5, -7 ], [ -6 ]>>
# IdealsTest11: greens.gi for ideals
gap> S := Semigroup([Transformation([1, 2, 6, 6, 5, 5]),
> Transformation([2, 3, 1, 3, 6, 5]),
> Transformation([3, 4, 6, 3, 2, 1]),
> Transformation([3, 6, 6, 4, 5, 6]),
> Transformation([4, 5, 2, 1, 6, 5]),
> Transformation([4, 6, 4, 2, 5, 2]),
> Transformation([5, 3, 2, 1, 2, 1]),
> Transformation([6, 5, 2, 4, 4, 1])]);
<transformation semigroup of degree 6 with 8 generators>
gap> gensI := [Transformation([4, 5, 2, 1, 6, 5]),
> Transformation([5, 2, 6, 2, 1, 4]),
> Transformation([3, 4, 6, 3, 2, 1])];;
gap> gensJ := [Transformation([1, 2, 6, 6, 5, 5])];;
gap> I := SemigroupIdeal(S, gensI);
<non-regular transformation semigroup ideal of degree 6 with 3 generators>
gap> J := SemigroupIdeal(S, gensJ);
<regular transformation semigroup ideal of degree 6 with 1 generator>
gap> T := Semigroup([PartialPerm([1, 2, 3, 4, 5], [1, 2, 6, 4, 5]),
> PartialPerm([2, 3, 4, 5, 6], [2, 1, 6, 4, 5]),
> PartialPerm([1, 2, 3, 4, 5], [3, 5, 4, 6, 2]),
> PartialPerm([1, 2, 4, 5, 6], [6, 2, 3, 1, 5]),
> PartialPerm([1, 3, 4, 5, 6], [2, 1, 4, 3, 5]),
> PartialPerm([1, 2, 5, 6], [4, 5, 2, 1]),
> PartialPerm([2, 4, 5, 6], [4, 6, 3, 2]),
> PartialPerm([2, 3, 4, 5], [6, 3, 4, 5]),
> PartialPerm([2, 3, 4, 6], [5, 3, 2, 4])]);
<partial perm semigroup of rank 6 with 9 generators>
gap> gensK := [
> PartialPerm([2, 5, 6], [6, 4, 5]),
> PartialPerm([1, 2, 4, 6], [2, 3, 6, 1]),
> PartialPerm([1, 2, 3, 5], [4, 5, 1, 2]),
> PartialPerm([1, 2, 5, 6], [5, 2, 6, 1])];;
gap> K := SemigroupIdeal(T, gensK);
<non-regular partial perm semigroup ideal of rank 6 with 4 generators>
gap> R := Semigroup([Bipartition([[1, 2, 3, 5, -3], [4, 6, 7, -5], [-1],
> [-2, -4, -7], [-6]]),
> Bipartition([[1, 2, 6, -5], [3], [4, 5, -2], [7, -1, -3, -4, -7],
> [-6]]),
> Bipartition([[1, 3, 4, 5, -2, -3, -5], [2, 6, -1, -6, -7], [7],
> [-4]]),
> Bipartition([[1, 3, 4, 6, -7], [2, 5, -1, -5], [7, -2, -3, -4],
> [-6]]),
> Bipartition([[1, 3], [2, 5, 6, -1, -2, -3], [4, 7, -4, -7], [-5],
> [-6]]), Bipartition([[1, 4, 5, -1, -2, -4, -6],
> [2, 3, 7, -3, -5, -7], [6]]),
> Bipartition([[1, -1, -4], [2, 3, 4, 5, 6, 7, -2, -6],
> [-3, -5, -7]]),
> Bipartition([[1, 7, -6], [2, 3, 4, 5, -1, -2, -4], [6, -3, -5],
> [-7]]),
> Bipartition([[1, 5, -2, -7], [2, 3, 6, -4], [4, -1, -5, -6], [7],
> [-3]]),
> Bipartition([[1, -3, -4], [2], [3, 7, -1, -7], [4, 5, -6], [6, -5],
> [-2]])]);
<bipartition semigroup of degree 7 with 10 generators>
gap> gensL :=
> [Bipartition([[1, 2, 3, 5, -3], [4, 6, 7, -5], [-1], [-2, -4, -7],
> [-6]]),
> Bipartition([[1, 2, 6, -5], [3], [4, 5, -2], [7, -1, -3, -4, -7],
> [-6]]),
> Bipartition([[1, 3], [2, 5, 6, -1, -2, -3], [4, 7, -4, -7], [-5],
> [-6]]),
> Bipartition([[1, -3, -4], [2], [3, 7, -1, -7], [4, 5, -6], [6, -5], [-2]])];;
gap> L := SemigroupIdeal(R, gensL);
<non-regular bipartition semigroup ideal of degree 7 with 4 generators>
gap> U := Semigroup([Transformation([1, 3, 2, 2, 4, 5]),
> Transformation([1, 5, 6, 6, 2, 5]),
> Transformation([1, 6, 4, 6, 3, 1]),
> Transformation([2, 1, 3, 3, 5, 2]),
> Transformation([2, 1, 5, 1, 2, 3]),
> Transformation([3, 3, 1, 5, 1, 1]),
> Transformation([4, 4, 4, 3, 3, 5]),
> Transformation([4, 4, 6, 3, 5, 6]),
> Transformation([5, 3, 3, 6, 2, 4]),
> Transformation([6, 4, 4, 6, 3, 2])]);
<transformation semigroup of degree 6 with 10 generators>
gap> A := SemigroupIdeal(U, Transformation([5, 3, 3, 6, 2, 4]));
<non-regular transformation semigroup ideal of degree 6 with 1 generator>
gap> B := SemigroupIdeal(U, Transformation([6, 4, 4, 6, 3, 2]));
<non-regular transformation semigroup ideal of degree 6 with 1 generator>
gap> C := SemigroupIdeal(U, Transformation([3, 3, 1, 5, 1, 1]));
<regular transformation semigroup ideal of degree 6 with 1 generator>
gap> ideals := [A, B, C, I, J, K, L];;
# IdealsTest12: GreensXClasses
gap> GreensDClasses(I);
[ <Green's D-class: Transformation( [ 4, 5, 2, 1, 6, 5 ] )>,
<Green's D-class: Transformation( [ 5, 2, 6, 2, 1, 4 ] )>,
<Green's D-class: Transformation( [ 3, 4, 6, 3, 2, 1 ] )>,
<Green's D-class: Transformation( [ 1, 6, 5, 4, 5, 6 ] )>,
<Green's D-class: Transformation( [ 5, 5, 1, 5, 5, 2 ] )>,
<Green's D-class: Transformation( [ 2, 2, 2, 2, 6, 2 ] )>,
<Green's D-class: Transformation( [ 5, 5, 5, 5, 5, 5 ] )>,
<Green's D-class: Transformation( [ 6, 5, 2, 5, 4, 1 ] )>,
<Green's D-class: Transformation( [ 2, 6, 5, 6, 1, 4 ] )>,
<Green's D-class: Transformation( [ 6, 5, 2, 5, 1, 4 ] )>,
<Green's D-class: Transformation( [ 2, 6, 5, 6, 4, 1 ] )>,
<Green's D-class: Transformation( [ 5, 2, 6, 2, 4, 1 ] )>,
<Green's D-class: Transformation( [ 3, 4, 6, 4, 2, 1 ] )>,
<Green's D-class: Transformation( [ 4, 6, 3, 6, 1, 2 ] )>,
<Green's D-class: Transformation( [ 3, 4, 6, 4, 1, 2 ] )>,
<Green's D-class: Transformation( [ 4, 6, 3, 6, 2, 1 ] )>,
<Green's D-class: Transformation( [ 6, 3, 4, 3, 1, 2 ] )>,
<Green's D-class: Transformation( [ 6, 3, 4, 3, 2, 1 ] )>,
<Green's D-class: Transformation( [ 2, 4, 1, 2 ] )>,
<Green's D-class: Transformation( [ 2, 4, 1, 4, 6, 5 ] )>,
<Green's D-class: Transformation( [ 2, 4, 1, 4 ] )>,
<Green's D-class: Transformation( [ 4, 1, 2, 1 ] )>,
<Green's D-class: Transformation( [ 4, 1, 2, 1, 6, 5 ] )>,
<Green's D-class: Transformation( [ 1, 2, 4, 2 ] )>,
<Green's D-class: Transformation( [ 1, 2, 4, 2, 6, 5 ] )> ]
gap> Number(GreensDClasses(I), IsRegularGreensClass);
4
gap> I := SemigroupIdeal(S, gensI);;
gap> NrRegularDClasses(I);
4
gap> GreensDClasses(J);
[ <Green's D-class: Transformation( [ 1, 2, 6, 6, 5, 5 ] )>,
<Green's D-class: Transformation( [ 1, 2, 5, 5, 5, 5 ] )>,
<Green's D-class: Transformation( [ 6, 5, 5, 5, 5, 5 ] )>,
<Green's D-class: Transformation( [ 6, 6, 6, 6, 6, 6 ] )> ]
gap> Number(GreensDClasses(J), IsRegularGreensClass);
4
gap> J := SemigroupIdeal(S, gensJ);;
gap> NrRegularDClasses(J);
4
gap> Size(GreensDClasses(K));
735
gap> Number(GreensDClasses(K), IsRegularGreensClass);
5
gap> K := SemigroupIdeal(T, gensK);;
gap> NrRegularDClasses(K);
5
gap> Size(GreensHClasses(I));
2347
gap> Size(GreensLClasses(I));
75
gap> GreensLClasses(I){[51 .. 60]};
[ <Green's L-class: Transformation( [ 3, 3, 5, 3, 3, 3 ] )>,
<Green's L-class: Transformation( [ 5, 5, 5, 5, 5, 5 ] )>,
<Green's L-class: Transformation( [ 6, 6, 6, 6, 6, 6 ] )>,
<Green's L-class: Transformation( [ 1, 1, 1, 1, 1, 1 ] )>,
<Green's L-class: Transformation( [ 2, 2, 2, 2, 2, 2 ] )>,
<Green's L-class: Transformation( [ 3, 3, 3, 3, 3, 3 ] )>,
<Green's L-class: Transformation( [ 4, 4, 4, 4, 4, 4 ] )>,
<Green's L-class: Transformation( [ 6, 5, 2, 5, 1, 4 ] )>,
<Green's L-class: Transformation( [ 2, 6, 5, 6, 1, 4 ] )>,
<Green's L-class: Transformation( [ 5, 2, 6, 2, 4, 1 ] )> ]
gap> Size(GreensRClasses(I));
156
gap> Size(GreensHClasses(J));
2326
gap> Size(GreensLClasses(J));
54
gap> Size(GreensRClasses(J));
135
gap> GreensRClasses(J){[97 .. 103]};
[ <Green's R-class: Transformation( [ 5, 1, 2, 2, 2, 5 ] )>,
<Green's R-class: Transformation( [ 5, 2, 1, 5, 5, 5 ] )>,
<Green's R-class: Transformation( [ 1, 2, 5, 2, 5, 5 ] )>,
<Green's R-class: Transformation( [ 2, 5, 1, 5, 2, 5 ] )>,
<Green's R-class: Transformation( [ 2, 1, 2, 5, 2, 1 ] )>,
<Green's R-class: Transformation( [ 1, 2, 5, 1, 1, 5 ] )>,
<Green's R-class: Transformation( [ 5, 2, 5, 2, 1, 5 ] )> ]
gap> Size(GreensHClasses(K));
1555
gap> GreensHClasses(K){[1337 .. 1342]};
[ <Green's H-class: [3,1](2)(4)(5)>, <Green's H-class: [3,2](1,4)(5)>,
<Green's H-class: [6,2](1,5,4)>, <Green's H-class: [3,4,5,1][6,2]>,
<Green's H-class: [3,4][6,2](1)(5)>, <Green's H-class: [3,4,5][6,1](2)> ]
gap> Size(GreensLClasses(K));
917
gap> Size(GreensRClasses(K));
791
# IdealsTest13: XClassReps
gap> DClassReps(J);
[ Transformation( [ 1, 2, 6, 6, 5, 5 ] ),
Transformation( [ 1, 2, 5, 5, 5, 5 ] ),
Transformation( [ 6, 5, 5, 5, 5, 5 ] ),
Transformation( [ 6, 6, 6, 6, 6, 6 ] ) ]
gap> LClassReps(L){[10 .. 20]};
[ <bipartition: [ 1, 2, 3, 4, 5, 6, 7, -2, -7 ], [ -1, -4, -5, -6 ], [ -3 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -1, -7 ], [ -2 ], [ -3, -4, -5, -6 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -1, -4, -5, -6 ], [ -2, -7 ], [ -3 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -3, -4, -5, -6 ], [ -1, -7 ], [ -2 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -1, -5 ], [ -2, -3, -4, -7 ], [ -6 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -2, -4, -7 ], [ -1, -5, -6 ], [ -3 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -2, -3, -5 ], [ -1, -6, -7 ], [ -4 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -2, -6 ], [ -1, -4 ], [ -3, -5, -7 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -5 ], [ -1, -3, -4, -6, -7 ], [ -2 ]>,
<block bijection: [ 1, 2, 3, 4, 5, 6, 7, -1, -2, -3, -4, -5, -6, -7 ]>,
<bipartition: [ 1, 2, 3, 4, 5, 6, 7, -1, -6, -7 ], [ -2, -3, -5 ], [ -4 ]> ]
# IdealsTest14: GreensXClassOfElement
gap> GreensDClassOfElement(I, Transformation([2, 2, 6, 2, 5, 5]));
<Green's D-class: Transformation( [ 2, 2, 6, 2, 5, 5 ] )>
gap> Transformation([5, 5, 2, 5, 5, 1]) in last;
true
gap> GreensDClassOfElement(J, Transformation([6, 6, 4, 2, 4, 2]));
<Green's D-class: Transformation( [ 6, 6, 4, 2, 4, 2 ] )>
gap> Transformation([2, 1, 5, 5, 5, 5]) in last;
true
gap> GreensDClassOfElement(K, PartialPerm([1, 4, 5, 6], [6, 2, 1, 3]));
<Green's D-class: [4,2][5,1,6,3]>
gap> GreensHClassOfElement(L, Bipartition([[1, 3, 4, 6, -3],
> [2, 5, 7], [-1], [-2, -4, -7], [-5], [-6]]));
<Green's H-class: <bipartition: [ 1, 3, 4, 6, -3 ], [ 2, 5, 7 ], [ -1 ],
[ -2, -4, -7 ], [ -5 ], [ -6 ]>>
gap> GreensLClassOfElement(A, Transformation([6, 3, 4, 4, 3, 2]));
<Green's L-class: Transformation( [ 6, 3, 4, 4, 3, 2 ] )>
gap> GreensRClassOfElement(B, Transformation([3, 4, 6, 6, 4, 2]));
<Green's R-class: Transformation( [ 3, 4, 6, 6, 4, 2 ] )>
gap> GreensHClassOfElement(C, Transformation([2, 2, 5, 2, 2, 1]));
<Green's H-class: Transformation( [ 2, 2, 5, 2, 2, 1 ] )>
# IdealsTest15: NrXClasses (Recreate objects)
gap> I := SemigroupIdeal(S, gensI);;
gap> J := SemigroupIdeal(S, gensJ);;
gap> K := SemigroupIdeal(T, gensK);;
gap> L := SemigroupIdeal(R, gensL);;
gap> A := SemigroupIdeal(U, Transformation([5, 3, 3, 6, 2, 4]));;
gap> B := SemigroupIdeal(U, Transformation([6, 4, 4, 6, 3, 2]));;
gap> C := SemigroupIdeal(U, Transformation([3, 3, 1, 5, 1, 1]));;
gap> ideals := [A, B, C, I, J, K, L];;
gap> List(ideals, NrDClasses);
[ 58, 269, 3, 25, 4, 735, 63 ]
gap> List(ideals, NrRClasses);
[ 623, 347, 81, 156, 135, 791, 120 ]
gap> List(ideals, NrLClasses);
[ 269, 307, 41, 75, 54, 917, 165 ]
gap> List(ideals, NrHClasses);
[ 2225, 1722, 1456, 2347, 2326, 1555, 1568 ]
# IdealsTest16: NrRegularDClasses
gap> List(ideals, NrRegularDClasses);
[ 4, 3, 3, 4, 4, 5, 4 ]
# IdealsTest17: Idempotents, (and with integer)
gap> L := SemigroupIdeal(R, gensL);;
gap> Idempotents(A){[444 .. 450]};
[ Transformation( [ 4, 4, 3, 4, 3, 3 ] ),
Transformation( [ 2, 2, 6, 2, 6, 6 ] ),
Transformation( [ 2, 2, 5, 2, 5, 5 ] ),
Transformation( [ 2, 2, 3, 2, 3, 3 ] ),
Transformation( [ 1, 1, 6, 1, 6, 6 ] ),
Transformation( [ 4, 4, 5, 4, 5, 5 ] ),
Transformation( [ 1, 1, 3, 1, 3, 3 ] ) ]
gap> Idempotents(B){[444 .. 450]};
[ Transformation( [ 1, 2, 6, 6, 6, 6 ] ),
Transformation( [ 1, 2, 5, 5, 5, 5 ] ),
Transformation( [ 1, 2, 3, 3, 3, 3 ] ),
Transformation( [ 1, 4, 4, 4, 6, 6 ] ),
Transformation( [ 1, 3, 3, 3, 6, 6 ] ),
Transformation( [ 1, 4, 4, 4, 5, 5 ] ),
Transformation( [ 1, 3, 3, 3, 5, 5 ] ) ]
gap> Idempotents(C){[444 .. 450]};
[ Transformation( [ 1, 6, 3, 6, 1, 6 ] ), Transformation( [ 1, 3, 3, 1, 3 ] ),
Transformation( [ 1, 3, 3, 1, 6, 6 ] ), Transformation( [ 1, 3, 3, 3, 1 ] ),
Transformation( [ 1, 3, 3, 3, 6, 6 ] ),
Transformation( [ 1, 3, 3, 6, 6, 6 ] ), Transformation( [ 1, 1, 3, 3, 3 ] )
]
gap> Idempotents(C, 1);
[ Transformation( [ 4, 4, 4, 4, 4, 4 ] ),
Transformation( [ 2, 2, 2, 2, 2, 2 ] ),
Transformation( [ 6, 6, 6, 6, 6, 6 ] ),
Transformation( [ 3, 3, 3, 3, 3, 3 ] ),
Transformation( [ 1, 1, 1, 1, 1, 1 ] ),
Transformation( [ 5, 5, 5, 5, 5, 5 ] ) ]
gap> Idempotents(L, 2);
[ <bipartition: [ 1, 3 ], [ 2, 5, 6, -1, -2, -3 ], [ 4, 7, -4, -7 ], [ -5 ],
[ -6 ]>,
<bipartition: [ 1, 7, -1, -7 ], [ 2, 3, 4, 5, 6, -3, -4, -5, -6 ], [ -2 ]>,
<bipartition: [ 1, 4, 5, 6, -1, -4, -5, -6 ], [ 2 ], [ 3, 7, -2, -7 ],
[ -3 ]>, <bipartition: [ 1, 3, 4, 5, 7, -1, -2, -3, -4, -5 ], [ 2 ],
[ 6, -6 ], [ -7 ]>,
<bipartition: [ 1, 3, 4, 6 ], [ 2, 5, -5 ], [ 7, -1, -3, -4, -6, -7 ],
[ -2 ]>, <bipartition: [ 1, 5, -1, -5 ], [ 2, 3, 4, 6, -2, -3, -4, -7 ],
[ 7 ], [ -6 ]> ]
# IdealsTest18: NrIdempotents
gap> List(ideals, NrIdempotents);
[ 547, 528, 528, 774, 774, 43, 1406 ]
gap> List(ideals, i -> Size(Idempotents(i)));
[ 547, 528, 528, 774, 774, 43, 1406 ]
# IdealsTest19: PartialOrderOfDClasses
gap> PartialOrderOfDClasses(K);
<immutable digraph with 735 vertices, 6707 edges>
gap> PartialOrderOfDClasses(I);
<immutable digraph with 25 vertices, 69 edges>
gap> PartialOrderOfDClasses(J);
<immutable digraph with 4 vertices, 5 edges>
gap> PartialOrderOfDClasses(A);
<immutable digraph with 58 vertices, 216 edges>
gap> PartialOrderOfDClasses(B);
<immutable digraph with 269 vertices, 801 edges>
gap> PartialOrderOfDClasses(C);
<immutable digraph with 3 vertices, 3 edges>
# IdealsTest20: Check that sizes were correct
gap> List(ideals, Size);
[ 9285, 7172, 6906, 19167, 19146, 3782, 1568 ]
# IdealsTest21
# this example caused a seg fault before changeset 34d25659aa72
gap> S := InverseSemigroup([
> PartialPerm([1, 2, 3, 4], [4, 1, 2, 6]),
> PartialPerm([1, 2, 3, 4], [5, 7, 1, 6]),
> PartialPerm([1, 2, 3, 5], [5, 2, 7, 3]),
> PartialPerm([1, 2, 3, 6, 7], [1, 3, 4, 7, 5]),
> PartialPerm([1, 2, 3, 4, 5, 7], [3, 2, 4, 6, 1, 5])]);
<inverse partial perm semigroup of rank 7 with 5 generators>
gap> I := SemigroupIdeal(S, PartialPerm([1, 3, 4, 5, 7], [1, 3, 4, 5, 7]));
<inverse partial perm semigroup ideal of rank 7 with 1 generator>
gap> GeneratorsOfSemigroup(I);;
# IdealsTest22
# test for \in method from idealact.gi (only applies to ideals that know
# apriori that they are regular) which partially enumerates the semigroup ideal
# data
gap> S := RegularSemigroup(FullTransformationMonoid(6));;
gap> x := Transformation([6, 5, 1, 5, 1, 2]);;
gap> I := SemigroupIdeal(S, x);
<regular transformation semigroup ideal of degree 6 with 1 generator>
gap> x := Transformation([2, 2, 4, 4, 2, 2]);;
gap> x in I;
true
gap> IsClosedData(SemigroupData(I));
false
gap> Size(I);
35136
gap> Size(SemigroupIdeal(S, I));
35136
# IdealsTest23: IsomorphismPermGroup
# for an ideal which happens to be a group...
gap> S := FullTransformationSemigroup(6);;
gap> S := Semigroup(GroupOfUnits(S));;
gap> I := SemigroupIdeal(S, S);;
gap> x := IsomorphismPermGroup(I);;
gap> Source(x) = I;
true
gap> Range(x);
Group([ (1,2,3,4,5,6), (1,2) ])
#
gap> SEMIGROUPS.StopTest();
gap> STOP_TEST("Semigroups package: extreme/ideals.tst");
