#############################################################################
##
#W extreme/closure.tst
#Y Copyright (C) 2011-15 James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
##
#@local S, T, a, acting, gens, i, s, t, x
gap> START_TEST("Semigroups package: extreme/closure.tst");
gap> LoadPackage("semigroups", false);;
#
gap> SEMIGROUPS.StartTest();
# ClosureTest1
gap> gens := [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> S := Monoid(gens[1]);; Size(S);;
gap> for i in [2 .. 4] do
> S := ClosureSemigroup(S, gens[i]); Size(S);
> od;
gap> Size(S);
233606
gap> NrRClasses(S);
4397
gap> NrLClasses(S);
16915
gap> NrDClasses(S);
662
gap> GroupOfUnits(S);
<trivial transformation group of degree 0 with 1 generator>
# ClosureTest2
gap> gens :=
> [Transformation([1, 3, 9, 3, 12, 1, 15, 1, 19, 3, 1, 9, 1, 9, 22, 15, 3, 1,
> 24, 15, 1, 26, 1, 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1,
> 44, 1, 46, 1, 48, 1, 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64,
> 1, 66, 1, 68, 1, 70, 1, 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84, 1,
> 86, 1, 88, 1, 90, 1, 92, 1, 94, 1, 96, 1, 98, 1, 100, 1, 102, 1, 104, 1,
> 106, 1, 108, 1, 110, 1, 112, 1, 114, 1, 116, 1, 118, 1, 120, 1, 122, 1,
> 124, 1, 125, 1, 1, 1, 1]),
> Transformation([1, 4, 1, 10, 10, 4, 1, 17, 1, 1, 10, 1, 4, 1, 1, 1, 16, 4,
> 1, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 1, 1]),
> Transformation([1, 5, 5, 11, 1, 1, 5, 1, 5, 11, 1, 5, 1, 5, 5, 11, 11, 1,
> 5, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 1, 5, 1]),
> Transformation([1, 6, 1, 6, 6, 13, 1, 18, 1, 6, 6, 1, 21, 1, 1, 1, 6, 23,
> 1, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43,
> 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1,
> 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1,
> 85, 1, 87, 1, 89, 1, 91, 1, 93, 1, 95, 1, 97, 1, 99, 1, 101, 1, 103, 1,
> 105, 1, 107, 1, 109, 1, 111, 1, 113, 1, 115, 1, 117, 1, 119, 1, 121, 1,
> 123, 1, 1, 1, 126, 1, 1, 1]),
> Transformation([1, 7, 7, 1, 1, 14, 16, 1, 7, 1, 1, 20, 14, 16, 7, 1, 1,
> 14, 7, 1, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14,
> 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 7, 14]),
> Transformation([1, 8, 8, 1, 1, 1, 8, 1, 8, 1, 1, 8, 1, 8, 8, 1, 1, 1, 8,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 8, 1])];;
gap> s := Semigroup(gens[1]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]);
> od;
gap> s;
<transformation semigroup of degree 126 with 6 generators>
gap> Size(s);
15853
gap> Size(Semigroup(Generators(s)));
15853
gap> NrRClasses(s);
355
gap> t := Semigroup(gens);
<transformation semigroup of degree 126 with 6 generators>
gap> NrRClasses(t);
355
gap> NrLClasses(s);
353
gap> NrLClasses(t);
353
# ClosureTest3
gap> s := Semigroup(gens[1]);; Size(s);
30
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 15853, degree 126 with 6 generators>
gap> Size(s);
15853
gap> Size(Semigroup(Generators(s)));
15853
gap> NrRClasses(s);
355
gap> t := Semigroup(gens);
<transformation semigroup of degree 126 with 6 generators>
gap> NrRClasses(t);
355
gap> NrLClasses(s);
353
gap> NrLClasses(t);
353
# ClosureTest4
gap> gens := [Transformation([11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]),
> Transformation([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10]),
> Transformation([2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10]),
> Transformation([4, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10]),
> Transformation([2, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10]),
> Transformation([1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9]),
> Transformation([4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8]),
> Transformation([6, 5, 4, 3, 2, 3, 4, 5, 6, 7, 8]),
> Transformation([1, 2, 3, 4, 5, 6, 5, 4, 5, 6, 7]),
> Transformation([4, 5, 6, 7, 8, 7, 6, 7, 8, 9, 10]),
> Transformation([4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4])];;
gap> s := Semigroup(gens[1]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 6996, degree 11 with 11 generators>
gap> Size(s);
6996
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
512
392
46
423
gap> t := Semigroup(gens);
<transformation semigroup of degree 11 with 11 generators>
gap> NrRClasses(t); NrLClasses(t); NrDClasses(t); NrIdempotents(t);
512
392
46
423
# ClosureTest5
gap> s := Semigroup(gens[1]);;
gap> s := ClosureSemigroup(s, gens[2]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 6996, degree 11 with 11 generators>
gap> Size(s);
6996
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
512
392
46
423
gap> t := Semigroup(gens);
<transformation semigroup of degree 11 with 11 generators>
gap> NrRClasses(t); NrLClasses(t); NrDClasses(t); NrIdempotents(t);
512
392
46
423
# ClosureTest6
gap> gens := [Transformation([3, 4, 1, 2, 1]),
> Transformation([4, 2, 1, 5, 5]),
> Transformation([4, 2, 2, 2, 4])];;
gap> s := Monoid(gens[1], gens[2]);
<transformation monoid of degree 5 with 2 generators>
gap> s := ClosureSemigroup(s, gens[3]);
<transformation monoid of degree 5 with 3 generators>
gap> Size(s);
732
gap> IsRegularSemigroup(s);
true
gap> MultiplicativeZero(s);
fail
gap> GroupOfUnits(s);
<trivial transformation group of degree 0 with 1 generator>
# ClosureTest8
gap> gens := [Transformation([1, 3, 4, 1]),
> Transformation([2, 4, 1, 2]),
> Transformation([3, 1, 1, 3]),
> Transformation([3, 3, 4, 1])];;
gap> s := Monoid(gens[3]);
<commutative transformation monoid of degree 4 with 1 generator>
gap> for i in [1 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation monoid of size 62, degree 4 with 4 generators>
gap> Size(s);
62
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
10
15
6
20
# ClosureTest10
gap> gens := [Transformation([1, 3, 2, 3]),
> Transformation([1, 4, 1, 2]),
> Transformation([2, 4, 1, 1]),
> Transformation([3, 4, 2, 2])];;
gap> S := Monoid(gens[1]);;
gap> for x in gens do
> S := ClosureSemigroup(S, x);
> od;
gap> Size(S);
115
gap> NrRClasses(S); NrLClasses(S); NrDClasses(S); NrIdempotents(S);
12
20
6
29
# ClosureTest11
gap> gens := [Transformation([1, 3, 2, 3]),
> Transformation([1, 4, 1, 2]),
> Transformation([3, 4, 2, 2]),
> Transformation([4, 1, 2, 1])];;
gap> s := Monoid(gens[1]);;
gap> for i in [1 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]);
> od;
gap> Size(s);
69
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
17
21
9
22
# ClosureSemigroup: testing for Froidure-Pin algorithm performance, 1/?
gap> S := InverseSemigroup(
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8], [2, 3, 4, 5, 6, 7, 1, 8]),
> PartialPerm([1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 7, 8, 6]),
> PartialPerm([1, 2, 3, 5, 6, 7, 8], [5, 6, 7, 1, 2, 3, 4]),
> PartialPerm([1, 2, 3, 4, 5, 6, 8], [2, 3, 4, 5, 6, 7, 1]),
> rec(acting := false));;
gap> Size(S);
1421569
gap> T := ClosureSemigroup(S, AsPartialPerm((1, 2), 8));
<partial perm semigroup of rank 8 with 7 generators>
gap> Size(T);
1441729
gap> IsEnumerated(T);
true
# ClosureSemigroup: testing for Froidure-Pin algorithm performance, 2/?
gap> gens :=
> [PBR([[], [-1]], [[2], [-2, 1]]),
> PBR([[-2, 1], [-1]], [[2], []]),
> PBR([[-1, 2], [-2]], [[1], [2]]),
> PBR([[-1], [-2]], [[1], [-2, 2]]),
> PBR([[-2], [2]], [[1], [2]]),
> PBR([[-2], [-1]], [[1], [1, 2]]),
> PBR([[-2], [-1]], [[1], [2]]),
> PBR([[-2], [-1]], [[1], [-2]]),
> PBR([[-2], [-1]], [[2], [1]]),
> PBR([[-2], [-2, -1]], [[1], [2]])];;
gap> S := Semigroup(gens[1]);;
gap> for i in [1 .. 10] do
> S := ClosureSemigroup(S, gens[i]);
> od;
gap> Size(S);
65536
gap> IsEnumerated(S);
true
# Adding redundant generators in ClosureSemigroups
gap> gens := [Transformation([10, 10, 6, 9, 3, 6, 6, 8, 3, 4]),
> Transformation([10, 10, 6, 9, 3, 6, 6, 8, 3, 4]),
> Transformation([6, 8, 8, 4, 7, 1, 2, 9, 9, 3]),
> Transformation([6, 8, 8, 4, 7, 1, 2, 9, 9, 3]),
> Transformation([9, 9, 1, 3, 4, 10, 5, 6, 3, 3]),
> Transformation([9, 9, 1, 3, 4, 10, 5, 6, 3, 3]),
> Transformation([7, 1, 2, 9, 3, 10, 3, 2, 5, 6])];;
gap> S := Semigroup(gens{[1, 2]});;
gap> for i in [2 .. 7] do
> S := ClosureSemigroup(S, gens[i]);
> Size(S);
> od;
gap> Size(S);
3063073
gap> S := Semigroup(gens{[1, 2]});; Size(S);
5
gap> S := ClosureSemigroup(S, gens{[3, 4]});; Size(S);
5662
gap> S := ClosureSemigroup(S, gens{[5, 6]});; Size(S);
111926
gap> S := ClosureSemigroup(S, gens{[7]});; Size(S);
3063073
#
gap> gens := [
> PartialPerm([2, 3, 4, 5, 6, 7, 1]),
> PartialPerm([2, 1, 0, 4, 0, 6, 0]),
> PartialPerm([1, 0, 0, 4, 5, 6, 0]),
> PartialPerm([7, 6, 2, 1, 0, 0, 0]),
> PartialPerm([7, 2, 3, 1, 0, 4, 0]),
> PartialPerm([6, 0, 1, 0, 0, 4, 7]),
> PartialPerm([7, 2, 3, 1, 0, 4, 0]),
> PartialPerm([4, 0, 3, 0, 6, 2, 0]),
> PartialPerm([4, 2, 3, 0, 0, 1, 0]),
> PartialPerm([1, 5, 2, 0, 6, 0, 0]),
> PartialPerm([1, 2, 0, 3, 0, 5, 0]),
> PartialPerm([2, 6, 0, 5, 0, 0, 4]),
> PartialPerm([2, 7, 3, 0, 4, 0, 0]),
> PartialPerm([7, 3, 4, 0, 0, 2, 0]),
> PartialPerm([2, 5, 3, 0, 0, 0, 4])];;
gap> S := Semigroup(gens[1]);;
gap> a := [];;
gap> for i in [1 .. Length(gens)] do
> S := ClosureSemigroup(S, gens[i]);
> Add(a, Size(S));
> od;
gap> a;
[ 7, 498, 743, 3977, 11229, 11817, 11817, 11915, 13679, 13826, 14414, 15002,
15198, 16766, 17354 ]
# testing Froidure-Pin closure semigroup for bipartitions
gap> gens := [Bipartition([[1, 4, -1], [2, -3], [3, 6, -5],
> [5, -2, -4, -6]]),
> Bipartition([[1, -1], [2, 5, 6, -2], [3, -4], [4, -3, -6],
> [-5]]),
> Bipartition([[1, 5], [2, -4], [3, 6, -1], [4, -5], [-2],
> [-3, -6]]),
> Bipartition([[1, 2, 3, -1, -5, -6], [4, 5, -2], [6, -3, -4]]),
> Bipartition([[1, 2, 3, -2], [4, -1], [5], [6, -6], [-3, -4],
> [-5]]),
> Bipartition([[1, 6, -2, -4, -6], [2, 3, 4, -1, -3], [5, -5]]),
> Bipartition([[1, 2, 3, -6], [4, 6, -3], [5], [-1, -2],
> [-4, -5]]),
> Bipartition([[1, 2, 4, -6], [3, 5, 6, -1], [-2, -3, -4, -5]]),
> Bipartition([[1, 2, 6, -1, -4, -5, -6], [3, 4, -2, -3], [5]]),
> Bipartition([[1, 5, 6, -5], [2, 3, 4], [-1], [-2, -3, -4],
> [-6]])];;
gap> a := [];; S := Semigroup(gens{[1 .. 4]});; Size(S);
2328
gap> for i in [1 .. Length(gens) / 2] do
> S := ClosureSemigroup(S, [gens[2 * i - 1], gens[2 * i]]);
> Add(a, Size(S));
> od;
gap> a;
[ 2328, 2328, 4257, 5274, 5555 ]
gap> a := [];; S := Semigroup(gens{[1 .. 4]});; Size(S);;
gap> for i in [1 .. Length(gens)] do
> S := ClosureSemigroup(S, gens[i]);
> Add(a, Size(S));
> od;
gap> a;
[ 2328, 2328, 2328, 2328, 3841, 4257, 4358, 5274, 5468, 5555 ]
# ClosureSemigroup for boolean matrices
gap> gens := [
> BooleanMat([[false, true, false, false], [true, false, false, false],
> [false, false, false, true], [true, true, false, true]]),
> BooleanMat([[true, true, true, true], [true, false, false, true],
> [true, false, false, true], [true, true, false, true]]),
> BooleanMat([[true, true, true, false], [false, false, true, true],
> [true, false, true, false], [false, false, true, true]]),
> BooleanMat([[true, true, false, false], [true, true, false, false],
> [false, true, false, false], [false, false, true, false]]),
> BooleanMat([[true, false, false, true], [false, false, false, false],
> [false, false, true, true], [true, false, true, false]]),
> BooleanMat([[false, true, true, true], [false, true, true, true],
> [true, false, false, false], [true, true, false, false]]),
> BooleanMat([[false, false, true, true], [true, false, true, true],
> [true, true, true, false], [false, true, false, true]]),
> BooleanMat([[false, false, true, false], [false, true, false, false],
> [true, false, false, false], [true, true, false, true]]),
> BooleanMat([[false, false, true, false], [false, true, false, false],
> [false, true, false, false], [false, true, false, true]]),
> BooleanMat([[false, false, false, true], [false, true, true, true],
> [true, true, true, false], [false, true, true, false]]),
> BooleanMat([[false, false, false, true], [false, false, false, true],
> [false, false, true, false], [true, false, false, false]]),
> BooleanMat([[true, false, false, true], [false, true, true, false],
> [true, false, true, true], [false, false, false, true]]),
> BooleanMat([[false, false, true, true], [true, false, true, true],
> [true, true, false, false], [false, true, true, false]]),
> BooleanMat([[true, true, false, false], [false, true, true, true],
> [true, true, true, false], [false, true, false, true]]),
> BooleanMat([[false, false, false, true], [true, true, true, true],
> [false, false, true, true], [false, true, false, true]]),
> BooleanMat([[true, true, true, true], [false, true, false, true],
> [false, false, true, true], [false, false, false, true]]),
> BooleanMat([[false, true, true, false], [true, false, true, false],
> [true, false, false, true], [false, false, false, true]])];;
gap> a := [];;S := Semigroup(gens{[1 .. 10]});; Size(S);;
gap> for i in [10 .. Length(gens)] do;
> S := ClosureSemigroup(S, gens[i]);
> Add(a, Size(S));
> od;
gap> a;
[ 2346, 3316, 3593, 3767, 4000, 4191, 4290, 4747 ]
# ClosureTest1
gap> gens := [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> s := Monoid(gens[1]);; Size(s);
5
gap> for i in [2 .. 4] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> Size(s);
233606
gap> NrRClasses(s);
4397
gap> NrLClasses(s);
16915
gap> NrDClasses(s);
662
gap> GroupOfUnits(s);
<trivial transformation group of degree 0 with 1 generator>
# ClosureTest2
gap> gens := [Transformation([
> 1, 3, 9, 3, 12, 1, 15, 1, 19, 3, 1, 9, 1, 9, 22, 15, 3, 1, 24, 15, 1, 26, 1,
> 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1,
> 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 1, 70, 1,
> 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84, 1, 86, 1, 88, 1, 90, 1, 92, 1,
> 94, 1, 96, 1, 98, 1, 100, 1, 102, 1, 104, 1, 106, 1, 108, 1, 110, 1, 112, 1,
> 114, 1, 116, 1, 118, 1, 120, 1, 122, 1, 124, 1, 125, 1, 1, 1, 1]),
> Transformation([1, 4, 1, 10, 10, 4, 1, 17, 1, 1, 10, 1, 4, 1, 1, 1, 16, 4,
> 1, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 1, 1]),
> Transformation([1, 5, 5, 11, 1, 1, 5, 1, 5, 11, 1, 5, 1, 5, 5, 11, 11, 1,
> 5, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5,
> 1, 5, 1, 5, 1, 5, 1, 5, 1, 1, 5, 1]),
> Transformation([1, 6, 1, 6, 6, 13, 1, 18, 1, 6, 6, 1, 21, 1, 1, 1, 6, 23,
> 1, 1, 25, 1, 27, 1, 29, 1, 31, 1, 33, 1, 35, 1, 37, 1, 39, 1, 41, 1, 43,
> 1, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 55, 1, 57, 1, 59, 1, 61, 1, 63, 1,
> 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 75, 1, 77, 1, 79, 1, 81, 1, 83, 1,
> 85, 1, 87, 1, 89, 1, 91, 1, 93, 1, 95, 1, 97, 1, 99, 1, 101, 1, 103, 1,
> 105, 1, 107, 1, 109, 1, 111, 1, 113, 1, 115, 1, 117, 1, 119, 1, 121, 1,
> 123, 1, 1, 1, 126, 1, 1, 1]),
> Transformation([1, 7, 7, 1, 1, 14, 16, 1, 7, 1, 1, 20, 14, 16, 7, 1, 1,
> 14, 7, 1, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14,
> 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7, 14, 7,
> 14, 7, 14, 7, 7, 14]),
> Transformation([1, 8, 8, 1, 1, 1, 8, 1, 8, 1, 1, 8, 1, 8, 8, 1, 1, 1, 8,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1,
> 8, 1, 8, 1, 8, 1, 8, 1, 8, 8, 1])];;
gap> s := Semigroup(gens[1]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]);
> od;
gap> s;
<transformation semigroup of degree 126 with 6 generators>
gap> Size(s);
15853
gap> Size(Semigroup(Generators(s)));
15853
gap> NrRClasses(s);
355
gap> t := Semigroup(gens);
<transformation semigroup of degree 126 with 6 generators>
gap> NrRClasses(t);
355
gap> NrLClasses(s);
353
gap> NrLClasses(t);
353
# ClosureTest3
gap> s := Semigroup(gens[1]);; Size(s);
30
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 15853, degree 126 with 6 generators>
gap> Size(s);
15853
gap> Size(Semigroup(Generators(s)));
15853
gap> NrRClasses(s);
355
gap> t := Semigroup(gens);
<transformation semigroup of degree 126 with 6 generators>
gap> NrRClasses(t);
355
gap> NrLClasses(s);
353
gap> NrLClasses(t);
353
# ClosureTest4
gap> gens := [Transformation([11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]),
> Transformation([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10]),
> Transformation([2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10]),
> Transformation([4, 3, 2, 3, 4, 5, 6, 7, 8, 9, 10]),
> Transformation([2, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10]),
> Transformation([1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9]),
> Transformation([4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8]),
> Transformation([6, 5, 4, 3, 2, 3, 4, 5, 6, 7, 8]),
> Transformation([1, 2, 3, 4, 5, 6, 5, 4, 5, 6, 7]),
> Transformation([4, 5, 6, 7, 8, 7, 6, 7, 8, 9, 10]),
> Transformation([4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4])];;
gap> s := Semigroup(gens[1]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 6996, degree 11 with 11 generators>
gap> Size(s);
6996
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
512
392
46
423
gap> t := Semigroup(gens);
<transformation semigroup of degree 11 with 11 generators>
gap> NrRClasses(t); NrLClasses(t); NrDClasses(t); NrIdempotents(t);
512
392
46
423
# ClosureTest5
gap> s := Semigroup(gens[1]);;
gap> s := ClosureSemigroup(s, gens[2]);;
gap> for i in [2 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation semigroup of size 6996, degree 11 with 11 generators>
gap> Size(s);
6996
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
512
392
46
423
gap> t := Semigroup(gens);
<transformation semigroup of degree 11 with 11 generators>
gap> NrRClasses(t); NrLClasses(t); NrDClasses(t); NrIdempotents(t);
512
392
46
423
# ClosureTest6
gap> gens := [Transformation([3, 4, 1, 2, 1]),
> Transformation([4, 2, 1, 5, 5]),
> Transformation([4, 2, 2, 2, 4])];;
gap> s := Monoid(gens[1], gens[2]);
<transformation monoid of degree 5 with 2 generators>
gap> s := ClosureSemigroup(s, gens[3]);
<transformation monoid of degree 5 with 3 generators>
gap> Size(s);
732
gap> IsRegularSemigroup(s);
true
gap> MultiplicativeZero(s);
fail
gap> GroupOfUnits(s);
<trivial transformation group of degree 0 with 1 generator>
# ClosureTest8
gap> gens := [Transformation([1, 3, 4, 1]),
> Transformation([2, 4, 1, 2]),
> Transformation([3, 1, 1, 3]),
> Transformation([3, 3, 4, 1])];;
gap> s := Monoid(gens[3]);
<commutative transformation monoid of degree 4 with 1 generator>
gap> for i in [1 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]); Size(s);
> od;
gap> s;
<transformation monoid of size 62, degree 4 with 4 generators>
gap> Size(s);
62
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
10
15
6
20
# ClosureTest10
gap> gens := [Transformation([1, 3, 2, 3]),
> Transformation([1, 4, 1, 2]),
> Transformation([2, 4, 1, 1]),
> Transformation([3, 4, 2, 2])];;
gap> s := Monoid(gens[1]);
<commutative transformation monoid of degree 4 with 1 generator>
gap> for i in [1 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]);
> od;
gap> Size(s);
115
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
12
20
6
29
# ClosureTest11
gap> gens := [Transformation([1, 3, 2, 3]),
> Transformation([1, 4, 1, 2]),
> Transformation([3, 4, 2, 2]),
> Transformation([4, 1, 2, 1])];;
gap> s := Monoid(gens[1]);
<commutative transformation monoid of degree 4 with 1 generator>
gap> for i in [1 .. Length(gens)] do
> s := ClosureSemigroup(s, gens[i]);
> od;
gap> Size(s);
69
gap> NrRClasses(s); NrLClasses(s); NrDClasses(s); NrIdempotents(s);
17
21
9
22
#
gap> SEMIGROUPS.StopTest();
gap> STOP_TEST("Semigroups package: extreme/closure.tst");
