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Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] #############################################################################
## #W standard/semigroups/semibipart.tst #Y Copyright (C) 2015-2022 James D. Mitchell ## ## Licensing information can be found in the README file of this package. ## ############################################################################# ## #@local A, B, BruteForceInverseCheck, BruteForceIsoCheck, C, F, R, S, T, an #@local gr1, gr2, inv, map, o1, o2, rels, x, y gap> START_TEST("Semigroups package: standard/semigroups/semibipart.tst"); gap> LoadPackage("semigroups", false);; # gap> SEMIGROUPS.StartTest(); # BruteForceIsoCheck helper functions gap> BruteForceIsoCheck := function(iso) > local x, y; > if not IsInjective(iso) or not IsSurjective(iso) then > return false; > fi; > for x in Generators(Source(iso)) do > for y in Generators(Source(iso)) do > if x ^ iso * y ^ iso <> (x * y) ^ iso then > return false; > fi; > od; > od; > return true; > end;; gap> BruteForceInverseCheck := function(map) > local inv; > inv := InverseGeneralMapping(map); > return ForAll(Source(map), x -> x = (x ^ map) ^ inv) > and ForAll(Range(map), x -> x = (x ^ inv) ^ map); > end;; # IsomorphismSemigroup for IsBlockBijection checks if the semigroup is inverse # inside the method and not in the filter to enter the method. gap> S := ReesMatrixSemigroup(Group([(1, 2)]), [[()]]); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> gap> S := AsSemigroup(IsBlockBijectionSemigroup, S); <block bijection group of size 2, degree 3 with 1 generator> gap> IsomorphismSemigroup(IsBlockBijectionSemigroup, > FullTransformationMonoid(2)); Error, the 2nd argument must be an inverse semigroup # AsSemigroup: # convert from IsPBRSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > PBR([[-2, 1, 2], [-2, 1, 2]], [[-1], [-2, 1, 2]]), > PBR([[-1, 1, 2], [-1, 1, 2]], [[-1, 1, 2], [-2]])]); <pbr semigroup of degree 2 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpSemigroup to IsBipartitionSemigroup gap> F := FreeSemigroup(2);; AssignGeneratorVariables(F);; gap> rels := [[s1 ^ 2, s1], [s1 * s2, s2], [s2 * s1, s1], [s2 ^ 2, s2]];; gap> S := F / rels; <fp semigroup with 2 generators and 4 relations of length 14> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Bipartition([[1, 2, -2], [-1]]), > Bipartition([[1, 2, -1], [-2]])]); <bipartition semigroup of degree 2 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of degree 2 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Transformation([2, 2]), Transformation([1, 1])]); <transformation semigroup of degree 2 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of degree 2 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsBooleanMat, > [[false, true], [false, true]]), > Matrix(IsBooleanMat, > [[true, false], [true, false]])]); <semigroup of 2x2 boolean matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of degree 2 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsMaxPlusMatrix, > [[-infinity, 0], [-infinity, 0]]), > Matrix(IsMaxPlusMatrix, > [[0, -infinity], [0, -infinity]])]); <semigroup of 2x2 max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsMinPlusMatrix, > [[infinity, 0], [infinity, 0]]), > Matrix(IsMinPlusMatrix, > [[0, infinity], [0, infinity]])]); <semigroup of 2x2 min-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, 0], [-infinity, 0]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[0, -infinity], [0, -infinity]])]); <semigroup of 2x2 projective max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(Integers, > [[0, 1], [0, 1]]), > Matrix(Integers, > [[1, 0], [1, 0]])]); <semigroup of 2x2 integer matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, 0], [-infinity, 0]], 5), > Matrix(IsTropicalMaxPlusMatrix, > [[0, -infinity], [0, -infinity]], 5)]); <semigroup of 2x2 tropical max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, 0], [infinity, 0]], 5), > Matrix(IsTropicalMinPlusMatrix, > [[0, infinity], [0, infinity]], 5)]); <semigroup of 2x2 tropical min-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixSemigroup to IsBipartitionSemigroup gap> S := Semigroup([ > Matrix(IsNTPMatrix, > [[0, 1], [0, 1]], 1, 4), > Matrix(IsNTPMatrix, > [[1, 0], [1, 0]], 1, 4)]); <semigroup of 2x2 ntp matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 2, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPBRMonoid to IsBipartitionSemigroup gap> S := Monoid([ > PBR([[-1, 1], [-2, 2], [-3, 3]], [[-1, 1], [-2, 2], [-3, 3]]), > PBR([[-3, -2, 1, 2, 3], [-3, -2, 1, 2, 3], [-3, -2, 1, 2, 3]], > [[-1], [-3, -2, 1, 2, 3], [-3, -2, 1, 2, 3]]), > PBR([[-3, 1, 2, 3], [-3, 1, 2, 3], [-3, 1, 2, 3]], > [[-2, -1], [-2, -1], [-3, 1, 2, 3]])]); <pbr monoid of degree 3 with 3 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 4, degree 4 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpMonoid to IsBipartitionSemigroup gap> F := FreeMonoid(3);; AssignGeneratorVariables(F);; gap> rels := [[m1 ^ 2, m1], [m1 * m2, m2], [m1 * m3, m3], [m2 * m1, m2], > [m2 ^ 2, m2], [m2 * m3, m3], [m3 * m1, m3], [m3 * m2, m2], [m3 ^ 2, m3]];; gap> S := F / rels; <fp monoid with 3 generators and 9 relations of length 30> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 4, degree 4 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Bipartition([[1, 2, 3, -2, -3], [-1]]), > Bipartition([[1, 2, 3, -3], [-1, -2]])]); <bipartition monoid of degree 3 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Transformation([2, 2, 2]), Transformation([3, 3, 3])]); <transformation monoid of degree 3 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Matrix(IsBooleanMat, > [[false, true, false], [false, true, false], [false, true, false]]), > Matrix(IsBooleanMat, > [[false, false, true], [false, false, true], [false, false, true]])]); <monoid of 3x3 boolean matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixMonoid to IsBipartitionSemigroup gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsMaxPlusMatrix, > [[x, 0, x], > [x, 0, x], > [x, 0, x]]), > Matrix(IsMaxPlusMatrix, > [[x, x, 0], > [x, x, 0], > [x, x, 0]])]); <monoid of 3x3 max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Matrix(IsMinPlusMatrix, > [[infinity, 0, infinity], > [infinity, 0, infinity], > [infinity, 0, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, 0], > [infinity, infinity, 0], > [infinity, infinity, 0]])]); <monoid of 3x3 min-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixMonoid to IsBipartitionSemigroup gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsProjectiveMaxPlusMatrix, > [[x, 0, x], > [x, 0, x], > [x, 0, x]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[x, x, 0], > [x, x, 0], > [x, x, 0]])]); <monoid of 3x3 projective max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Matrix(Integers, > [[0, 1, 0], [0, 1, 0], [0, 1, 0]]), > Matrix(Integers, > [[0, 0, 1], [0, 0, 1], [0, 0, 1]])]); <monoid of 3x3 integer matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixMonoid to IsBipartitionSemigroup gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsTropicalMaxPlusMatrix, > [[x, 0, x], > [x, 0, x], > [x, 0, x]], 2), > Matrix(IsTropicalMaxPlusMatrix, > [[x, x, 0], > [x, x, 0], > [x, x, 0]], 2)]); <monoid of 3x3 tropical max-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, 0, infinity], > [infinity, 0, infinity], > [infinity, 0, infinity]], 5), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, 0], > [infinity, infinity, 0], > [infinity, infinity, 0]], 5)]); <monoid of 3x3 tropical min-plus matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixMonoid to IsBipartitionSemigroup gap> S := Monoid([ > Matrix(IsNTPMatrix, > [[0, 1, 0], [0, 1, 0], [0, 1, 0]], 1, 5), > Matrix(IsNTPMatrix, > [[0, 0, 1], [0, 0, 1], [0, 0, 1]], 1, 5)]); <monoid of 3x3 ntp matrices with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsPBRSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > PBR([[-2, 1], [-1, 2], [-3, 3], [-4, 4]], > [[-1, 2], [-2, 1], [-3, 3], [-4, 4]]), > PBR([[1, 3], [-3, 2, 4], [1, 3], [-3, 2, 4]], > [[-4, -2, -1], [-4, -2, -1], [-3, 2, 4], [-4, -2, -1]]), > PBR([[-4, -3, -2, -1, 1, 2, 3, 4], [-4, -3, -2, -1, 1, 2, 3, 4], > [-4, -3, -2, -1, 1, 2, 3, 4], [-4, -3, -2, -1, 1, 2, 3, 4]], > [[-4, -3, -2, -1, 1, 2, 3, 4], [-4, -3, -2, -1, 1, 2, 3, 4], > [-4, -3, -2, -1, 1, 2, 3, 4], [-4, -3, -2, -1, 1, 2, 3, 4]])]); <pbr semigroup of degree 4 with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsFpSemigroup to IsBipartitionMonoid gap> F := FreeSemigroup(3);; AssignGeneratorVariables(F);; gap> rels := [[s1 * s3, s3], [s2 * s1, s2], [s2 ^ 2, s2], [s3 * s1, s3], > [s3 ^ 2, s3], [s1 ^ 3, s1], [s1 ^ 2 * s2, s2], [s2 * s3 * s2, s2], > [s3 * s2 * s3, s3]];; gap> S := F / rels; <fp semigroup with 3 generators and 9 relations of length 34> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsBipartitionMonoid to IsBipartitionMonoid gap> S := Semigroup([ > Bipartition([[1, -2], [2, -1], [3, -3], [4, -4]]), > Bipartition([[1, 3], [2, 4, -3], [-1, -2, -4]]), > Bipartition([[1, 2, 3, 4, -1, -2, -3, -4]])]); <bipartition semigroup of degree 4 with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTransformationSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Transformation([4, 2, 3, 1]), > Transformation([5, 2, 7, 2, 5, 2, 7, 5]), > Transformation([3, 6, 3, 3, 8, 6, 3, 8])]); <transformation semigroup of degree 8 with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsBooleanMatSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Matrix(IsBooleanMat, > [[false, false, false, true, false, false, false, false], > [false, true, false, false, false, false, false, false], > [false, false, true, false, false, false, false, false], > [true, false, false, false, false, false, false, false], > [false, false, false, false, true, false, false, false], > [false, false, false, false, false, true, false, false], > [false, false, false, false, false, false, true, false], > [false, false, false, false, false, false, false, true]]), > Matrix(IsBooleanMat, > [[false, false, false, false, true, false, false, false], > [false, true, false, false, false, false, false, false], > [false, false, false, false, false, false, true, false], > [false, true, false, false, false, false, false, false], > [false, false, false, false, true, false, false, false], > [false, true, false, false, false, false, false, false], > [false, false, false, false, false, false, true, false], > [false, false, false, false, true, false, false, false]]), > Matrix(IsBooleanMat, > [[false, false, true, false, false, false, false, false], > [false, false, false, false, false, true, false, false], > [false, false, true, false, false, false, false, false], > [false, false, true, false, false, false, false, false], > [false, false, false, false, false, false, false, true], > [false, false, false, false, false, true, false, false], > [false, false, true, false, false, false, false, false], > [false, false, false, false, false, false, false, true]])]); <semigroup of 8x8 boolean matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsMaxPlusMatrixSemigroup to IsBipartitionMonoid gap> x := -infinity;; gap> S := Semigroup([ > Matrix(IsMaxPlusMatrix, > [[x, x, x, 0, x, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [0, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0]]), > Matrix(IsMaxPlusMatrix, > [[x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, 0, x, x, x]]), > Matrix(IsMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0]])]); <semigroup of 8x8 max-plus matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsMinPlusMatrixSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Matrix(IsMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0]])]); <semigroup of 8x8 min-plus matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsProjectiveMaxPlusMatrixSemigroup to IsBipartitionMonoid gap> x := -infinity;; gap> S := Semigroup([ > Matrix(IsProjectiveMaxPlusMatrix, > [[x, x, x, 0, x, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [0, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, 0, x, x, x]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0]])]); <semigroup of 8x8 projective max-plus matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsIntegerMatrixSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Matrix(Integers, > [[0, 0, 0, 1, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 1]]), > Matrix(Integers, > [[0, 0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 1, 0, 0, 0]]), > Matrix(Integers, > [[0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1]])]); <semigroup of 8x8 integer matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTropicalMaxPlusMatrixSemigroup to IsBipartitionMonoid gap> x := -infinity;; gap> S := Semigroup([ > Matrix(IsTropicalMaxPlusMatrix, > [[x, x, x, 0, x, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [0, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0]], 5), > Matrix(IsTropicalMaxPlusMatrix, > [[x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x], > [x, 0, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x], > [x, x, x, x, 0, x, x, x]], 5), > Matrix(IsTropicalMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0], > [x, x, x, x, x, 0, x, x], > [x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, 0]], 5)]); <semigroup of 8x8 tropical max-plus matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTropicalMinPlusMatrixSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0]], > 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity, infinity]], > 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0], > [infinity, infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, infinity, infinity, 0]], > 3)]); <semigroup of 8x8 tropical min-plus matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsNTPMatrixSemigroup to IsBipartitionMonoid gap> S := Semigroup([ > Matrix(IsNTPMatrix, > [[0, 0, 0, 1, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 1]], 3, 4), > Matrix(IsNTPMatrix, > [[0, 0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 1, 0, 0, 0]], 3, 4), > Matrix(IsNTPMatrix, > [[0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1], > [0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1]], 3, 4)]); <semigroup of 8x8 ntp matrices with 3 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 8, degree 8 with 3 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsPBRMonoid to IsBipartitionMonoid gap> S := Monoid([ > PBR([[-2], [-2], [-5], [-7], [-5], [-9], [-7], [-10], [-9], [-10], > [-12], [-12]], > [[], [1, 2], [], [], [3, 5], [], [4, 7], [], [6, 9], [8, 10], [], > [11, 12]]), > PBR([[-3], [-4], [-6], [-4], [-8], [-6], [-4], [-8], [-11], [-8], [-11], > [-11]], > [[], [], [1], [2, 4, 7], [], [3, 6], [], [5, 8, 10], [], [], [9, 11, 12], > []])]); <pbr monoid of degree 12 with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsFpMonoid to IsBipartitionMonoid gap> F := FreeMonoid(2);; AssignGeneratorVariables(F);; gap> rels := [[m1 ^ 2, m1], > [m1 * m2 ^ 2, m1 * m2], > [m2 ^ 3, m2 ^ 2], > [(m1 * m2) ^ 2, m1 * m2]];; gap> S := F / rels; <fp monoid with 2 generators and 4 relations of length 21> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsBipartitionMonoid to IsBipartitionMonoid gap> S := Monoid([ > Bipartition([[1, 2, -2], [3], [4], [-1, -3], [-4]]), > Bipartition([[1, 4, -4], [2], [3, -1, -2, -3]])]); <bipartition monoid of degree 4 with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of degree 4 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTransformationMonoid to IsBipartitionMonoid gap> S := Monoid([ > Transformation([2, 2, 5, 7, 5, 9, 7, 10, 9, 10, 12, 12]), > Transformation([3, 4, 6, 4, 8, 6, 4, 8, 11, 8, 11, 11])]); <transformation monoid of degree 12 with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsBooleanMatMonoid to IsBipartitionMonoid gap> S := Monoid([ > Matrix(IsBooleanMat, > [[false, true, false, false, false, false, false, false, false, false, > false, false], > [false, true, false, false, false, false, false, false, false, false, > false, false], > [false, false, false, false, true, false, false, false, false, false, > false, false], > [false, false, false, false, false, false, true, false, false, false, > false, false], > [false, false, false, false, true, false, false, false, false, false, > false, false], > [false, false, false, false, false, false, false, false, true, false, > false, false], > [false, false, false, false, false, false, true, false, false, false, > false, false], > [false, false, false, false, false, false, false, false, false, true, > false, false], > [false, false, false, false, false, false, false, false, true, false, > false, false], > [false, false, false, false, false, false, false, false, false, true, > false, false], > [false, false, false, false, false, false, false, false, false, false, > false, true], > [false, false, false, false, false, false, false, false, false, false, > false, true]]), > Matrix(IsBooleanMat, > [[false, false, true, false, false, false, false, false, false, false, > false, false], > [false, false, false, true, false, false, false, false, false, false, > false, false], > [false, false, false, false, false, true, false, false, false, false, > false, false], > [false, false, false, true, false, false, false, false, false, false, > false, false], > [false, false, false, false, false, false, false, true, false, false, > false, false], > [false, false, false, false, false, true, false, false, false, false, > false, false], > [false, false, false, true, false, false, false, false, false, false, > false, false], > [false, false, false, false, false, false, false, true, false, false, > false, false], > [false, false, false, false, false, false, false, false, false, false, > true, false], > [false, false, false, false, false, false, false, true, false, false, > false, false], > [false, false, false, false, false, false, false, false, false, false, > true, false], > [false, false, false, false, false, false, false, false, false, false, > true, false]])]); <monoid of 12x12 boolean matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsMaxPlusMatrixMonoid to IsBipartitionMonoid gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsMaxPlusMatrix, > [[x, 0, x, x, x, x, x, x, x, x, x, x], > [x, 0, x, x, x, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, x, x, x, 0], > [x, x, x, x, x, x, x, x, x, x, x, 0]]), > Matrix(IsMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, x, x, x, 0, x]])]); <monoid of 12x12 max-plus matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsMinPlusMatrixMonoid to IsBipartitionMonoid gap> x := infinity;; gap> S := Monoid([ > Matrix(IsMinPlusMatrix, > [[x, 0, x, x, x, x, x, x, x, x, x, x], > [x, 0, x, x, x, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, x, x, x, 0], > [x, x, x, x, x, x, x, x, x, x, x, 0]]), > Matrix(IsMinPlusMatrix, > [[x, x, 0, x, x, x, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, x, x, x, 0, x]])]); <monoid of 12x12 min-plus matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsProjectiveMaxPlusMatrixMonoid to IsBipartitionMonoid gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsProjectiveMaxPlusMatrix, > [[x, 0, x, x, x, x, x, x, x, x, x, x], > [x, 0, x, x, x, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, x, x, x, 0], > [x, x, x, x, x, x, x, x, x, x, x, 0]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, x, x, x, 0, x]])]); <monoid of 12x12 projective max-plus matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsIntegerMatrixMonoid to IsBipartitionMonoid gap> S := Monoid([ > Matrix(Integers, > [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]), > Matrix(Integers, > [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]])]); <monoid of 12x12 integer matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTropicalMaxPlusMatrixMonoid to IsBipartitionMonoid gap> x := -infinity;; gap> S := Monoid([ > Matrix(IsTropicalMaxPlusMatrix, > [[x, 0, x, x, x, x, x, x, x, x, x, x], > [x, 0, x, x, x, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, x, x, x, 0], > [x, x, x, x, x, x, x, x, x, x, x, 0]], 5), > Matrix(IsTropicalMaxPlusMatrix, > [[x, x, 0, x, x, x, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, x, x, x, 0, x]], 5)]); <monoid of 12x12 tropical max-plus matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsTropicalMinPlusMatrixMonoid to IsBipartitionMonoid gap> x := infinity;; gap> S := Monoid([ > Matrix(IsTropicalMinPlusMatrix, > [[x, 0, x, x, x, x, x, x, x, x, x, x], > [x, 0, x, x, x, x, x, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, 0, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, 0, x, x, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, 0, x, x, x], > [x, x, x, x, x, x, x, x, x, 0, x, x], > [x, x, x, x, x, x, x, x, x, x, x, 0], > [x, x, x, x, x, x, x, x, x, x, x, 0]], 2), > Matrix(IsTropicalMinPlusMatrix, > [[x, x, 0, x, x, x, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, 0, x, x, x, x, x, x], > [x, x, x, 0, x, x, x, x, x, x, x, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, 0, x, x, x, x], > [x, x, x, x, x, x, x, x, x, x, 0, x], > [x, x, x, x, x, x, x, x, x, x, 0, x]], 2)]); <monoid of 12x12 tropical min-plus matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid: # convert from IsNTPMatrixMonoid to IsBipartitionMonoid gap> S := Monoid([ > Matrix(IsNTPMatrix, > [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]], 5, 5), > Matrix(IsNTPMatrix, > [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]], 5, 5)]); <monoid of 12x12 ntp matrices with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <bipartition monoid of size 12, degree 12 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesMatrixSemigroup to IsBipartitionSemigroup gap> R := ReesMatrixSemigroup(Group([(1, 2)]), [[()]]); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsSemigroup(IsBipartitionSemigroup, R); <block bijection group of size 2, degree 2 with 1 generator> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid # convert from IsReesMatrixSemigroup to IsBipartitionMonoid gap> R := ReesMatrixSemigroup(Group([(1, 2)]), [[(1, 2)]]); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsMonoid(IsBipartitionMonoid, R); <block bijection group of size 2, degree 2 with 1 generator> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesZeroMatrixSemigroup to IsBipartitionSemigroup gap> R := ReesZeroMatrixSemigroup(Group([(1, 2)]), > [[(1, 2)]]); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsSemigroup(IsBipartitionSemigroup, R); <bipartition semigroup of size 3, degree 3 with 2 generators> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid # convert from IsReesZeroMatrixSemigroup to IsBipartitionMonoid gap> R := ReesZeroMatrixSemigroup(Group([(1, 2)]), [[()]]); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsMonoid(IsBipartitionMonoid, R); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from graph inverse to IsBipartitionSemigroup gap> S := GraphInverseSemigroup(Digraph([[2], []])); <finite graph inverse semigroup with 2 vertices, 1 edge> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 6, degree 7 with 4 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionSemigroup to IsBipartitionSemigroup gap> S := InverseSemigroup(Bipartition([[1, -1, -3], [2, 3, -2]]));; gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse block bijection semigroup of degree 3 with 1 generator> gap> IsInverseSemigroup(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionMonoid to IsBipartitionMonoid gap> S := InverseMonoid([ > Bipartition([[1, -1, -3], [2, 3, -2]])]);; gap> T := AsMonoid(IsBipartitionMonoid, S); <inverse block bijection monoid of degree 3 with 1 generator> gap> IsInverseMonoid(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionMonoid to IsBipartitionSemigroup gap> S := InverseMonoid([ > Bipartition([[1, -1, -3], [2, 3, -2]])]);; gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse block bijection monoid of degree 3 with 1 generator> gap> IsInverseSemigroup(T) and IsMonoidAsSemigroup(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermSemigroup to IsBlockBijectionSemigroup # for an inverse semigroup gap> S := SymmetricInverseMonoid(2);; gap> T := AsSemigroup(IsBlockBijectionSemigroup, S); <inverse block bijection monoid of degree 3 with 2 generators> gap> IsInverseSemigroup(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBlockBijectionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermSemigroup to IsBlockBijectionSemigroup # for a non-inverse semigroup gap> S := Semigroup([ > PartialPerm([1, 2, 4], [4, 2, 3]), > PartialPerm([1, 3, 4], [1, 4, 3]), > PartialPerm([1, 2, 3, 4], [3, 4, 1, 2]), > PartialPerm([1, 3, 4], [2, 4, 1])]);; gap> T := AsSemigroup(IsBlockBijectionSemigroup, S); <block bijection semigroup of degree 5 with 4 generators> gap> IsInverseSemigroup(T); false gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBlockBijectionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesMatrixSemigroup to IsBipartitionSemigroup gap> R := ReesMatrixSemigroup(Group([(1, 2)]), [[(1, 2), (1, 2)], [(), ()]]); <Rees matrix semigroup 2x2 over Group([ (1,2) ])> gap> T := AsSemigroup(IsBipartitionSemigroup, R); <bipartition semigroup of size 8, degree 9 with 2 generators> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid # convert from IsReesMatrixSemigroup to IsBipartitionMonoid gap> R := ReesMatrixSemigroup(Group([(1, 2)]), [[(1, 2)]]); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsMonoid(IsBipartitionMonoid, R); <block bijection group of size 2, degree 2 with 1 generator> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesZeroMatrixSemigroup to IsBipartitionSemigroup gap> R := ReesZeroMatrixSemigroup(Group([(1, 2)]), > [[(1, 2), (1, 2)], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> T := AsSemigroup(IsBipartitionSemigroup, R); <bipartition semigroup of size 9, degree 10 with 3 generators> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsMonoid # convert from IsReesZeroMatrixSemigroup to IsBipartitionMonoid gap> R := ReesZeroMatrixSemigroup(Group([(1, 2)]), [[(1, 2)]]); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> gap> T := AsMonoid(IsBipartitionMonoid, R); <bipartition monoid of size 3, degree 3 with 2 generators> gap> Size(R) = Size(T); true gap> NrDClasses(R) = NrDClasses(T); true gap> NrRClasses(R) = NrRClasses(T); true gap> NrLClasses(R) = NrLClasses(T); true gap> NrIdempotents(R) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, R);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from graph inverse to IsBipartitionSemigroup gap> S := GraphInverseSemigroup(Digraph([[2], []])); <finite graph inverse semigroup with 2 vertices, 1 edge> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 6, degree 7 with 4 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from free band to IsBipartitionSemigroup gap> S := FreeBand(2); <free band on the generators [ x1, x2 ]> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of size 6, degree 7 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from perm group to IsBipartitionSemigroup gap> S := DihedralGroup(IsPermGroup, 6); Group([ (1,2,3), (2,3) ]) gap> T := AsSemigroup(IsBipartitionSemigroup, S); <block bijection group of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from perm group to IsBipartitionMonoid gap> S := DihedralGroup(IsPermGroup, 6); Group([ (1,2,3), (2,3) ]) gap> T := AsMonoid(IsBipartitionMonoid, S); <block bijection group of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from non-perm group to IsBipartitionSemigroup gap> S := DihedralGroup(6); <pc group of size 6 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <block bijection group of size 6, degree 6 with 5 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from non-perm group to IsBipartitionMonoid gap> S := DihedralGroup(6); <pc group of size 6 with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <block bijection group of size 6, degree 6 with 5 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionSemigroup to IsBipartitionSemigroup gap> S := InverseSemigroup(Bipartition([[1, -1, -3], [2, 3, -2]]));; gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse block bijection semigroup of degree 3 with 1 generator> gap> IsInverseSemigroup(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionMonoid to IsBipartitionMonoid gap> S := InverseMonoid([ > Bipartition([[1, -1, -3], [2, 3, -2]])]);; gap> T := AsMonoid(IsBipartitionMonoid, S); <inverse block bijection monoid of degree 3 with 1 generator> gap> IsInverseMonoid(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBlockBijectionMonoid to IsBipartitionSemigroup gap> S := InverseMonoid([ > Bipartition([[1, -1, -3], [2, 3, -2]])]);; gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse block bijection monoid of degree 3 with 1 generator> gap> IsInverseSemigroup(T) and IsMonoidAsSemigroup(T); true gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermSemigroup to IsBipartitionSemigroup gap> S := InverseSemigroup(PartialPerm([1, 2], [2, 1]), > PartialPerm([1, 2], [3, 1])); <inverse partial perm semigroup of rank 3 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse bipartition semigroup of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true gap> S := Semigroup(PartialPerm([1, 2], [2, 1]), > PartialPerm([1, 2], [3, 1])); <partial perm semigroup of rank 2 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <bipartition semigroup of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermMonoid to IsBipartitionMonoid gap> S := InverseMonoid(PartialPerm([1, 2], [2, 1]), > PartialPerm([1, 2], [3, 1])); <inverse partial perm monoid of rank 3 with 2 generators> gap> T := AsMonoid(IsBipartitionMonoid, S); <inverse bipartition monoid of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismMonoid(IsBipartitionMonoid, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermMonoid to IsBipartitionSemigroup gap> S := InverseMonoid(PartialPerm([1, 2], [2, 1]), > PartialPerm([1, 2], [3, 1])); <inverse partial perm monoid of rank 3 with 2 generators> gap> T := AsSemigroup(IsBipartitionSemigroup, S); <inverse bipartition monoid of degree 3 with 2 generators> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from an ideal to IsBipartitionSemigroup gap> S := SingularTransformationMonoid(3);; gap> T := AsSemigroup(IsBipartitionSemigroup, S);; gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBipartitionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from an ideal to IsBlockBijectionSemigroup gap> S := SymmetricInverseMonoid(3);; gap> S := SemigroupIdeal(S, S.3);; gap> T := AsSemigroup(IsBlockBijectionSemigroup, S); <inverse bipartition semigroup ideal of degree 4 with 1 generator> gap> Size(S) = Size(T); true gap> NrDClasses(S) = NrDClasses(T); true gap> NrRClasses(S) = NrRClasses(T); true gap> NrLClasses(S) = NrLClasses(T); true gap> NrIdempotents(S) = NrIdempotents(T); true gap> map := IsomorphismSemigroup(IsBlockBijectionSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # Test IsomorphismSemigroup # for IsBlockBijectionSemigroup and IsBlockBijectionSemigroup gap> S := DualSymmetricInverseMonoid(3); <inverse block bijection monoid of degree 3 with 3 generators> gap> map := IsomorphismSemigroup(IsBlockBijectionSemigroup, S); <inverse block bijection monoid of degree 3 with 3 generators> -> <inverse block bijection monoid of degree 3 with 3 generators> # Test RandomSemigroup gap> S := RandomSemigroup(IsBipartitionSemigroup, 2, 5);; gap> IsBipartitionSemigroup(S); true gap> DegreeOfBipartitionSemigroup(S); 5 gap> Length(GeneratorsOfSemigroup(S)); 2 gap> S := RandomSemigroup(IsBlockBijectionSemigroup, 2, 5);; gap> IsBlockBijectionSemigroup(S); true gap> DegreeOfBipartitionSemigroup(S); 5 gap> Length(GeneratorsOfSemigroup(S)); 2 # Test RandomMonoid gap> S := RandomMonoid(IsBipartitionMonoid, 4, 3);; gap> DegreeOfBipartitionSemigroup(S); 3 gap> Length(GeneratorsOfMonoid(S)); 4 gap> S := RandomMonoid(IsBlockBijectionMonoid, 2, 5);; gap> IsBlockBijectionMonoid(S); true gap> DegreeOfBipartitionSemigroup(S); 5 gap> Length(GeneratorsOfMonoid(S)); 2 # Test RandomInverseSemigroup gap> S := RandomInverseSemigroup(IsBlockBijectionSemigroup, 2, 5);; gap> IsBlockBijectionSemigroup(S); true gap> IsInverseSemigroup(S); true gap> DegreeOfBipartitionSemigroup(S); 5 gap> Length(GeneratorsOfInverseSemigroup(S)); 2 # Test RandomInverseMonoid gap> S := RandomInverseMonoid(IsBlockBijectionMonoid, 2, 5);; gap> IsBlockBijectionMonoid(S); true gap> IsInverseMonoid(S); true gap> DegreeOfBipartitionSemigroup(S); 5 gap> Length(GeneratorsOfInverseMonoid(S)); 2 # Test SemigroupViewStringPrefix gap> S := Monoid([ > Bipartition([[1, 3], [2, -2], [-1], [-3]]), > Bipartition([[1, -1], [2, -2], [3, -3]]), > Bipartition([[1, -1], [2, -2], [3], [-3]]), > Bipartition([[1], [2, -2], [3], [-1, -3]])]);; gap> IsStarSemigroup(S); true gap> SemigroupViewStringPrefix(S); "\>bipartition\< *-" gap> S := Semigroup(DualSymmetricInverseMonoid(4));; gap> IsStarSemigroup(S); true gap> SemigroupViewStringPrefix(S); "\>block bijection\< *-" # Test GroupOfUnits gap> S := Semigroup( > [Bipartition([[1, 5, 6, 10, 11, 12, 25, -3, -6, -16, -26, -27], > [2, 3, 4, 7, 13, 16, 17, 21, -2, -4, -13, -22, -24], > [8, 14, 18, -5, -7, -10, -19, -21], > [9, 15, 19, 20, 22, -1, -9, -14, -17, -23], > [23, 24, 26, 27, -8, -11, -18, -20], [-12, -15, -25]]), > Bipartition([[1, -1], [2, -2], [3, -3], [4, -4], [5, -5], > [6, -6], [7, -7], [8, -8], [9, -9], [10, -10], [11, -11], > [12, -12], [13, -14], [14, -16], [15, -17], [16, -19], > [17, -20], [18, -21], [19, -22], [20, -23], [21, -24], > [22, -13], [23, -25], [24, -26], [25, -15], [26, -27], > [27, -18]]), > Bipartition([[1, -2], [2, -3], [3, -5], > [4, -10], [5, -1], [6, -11], [7, -12], [8, -6], [9, -8], > [10, -7], [11, -9], [12, -4], [13, -13], [14, -14], > [15, -18], [16, -16], [17, -21], [18, -15], [19, -19], > [20, -24], [21, -17], [22, -22], [23, -26], [24, -20], > [25, -27], [26, -23], [27, -25]]), > Bipartition([[1, -4], [2, -6], [3, -7], [4, -8], [5, -9], > [6, -10], [7, -11], [8, -1], [9, -12], [10, -2], [11, -3], > [12, -5], [13, -15], [14, -17], [15, -18], [16, -20], > [17, -21], [18, -13], [19, -23], [20, -24], [21, -14], > [22, -25], [23, -26], [24, -16], [25, -27], [26, -19], > [27, -22]])]);; gap> GroupOfUnits(S); <block bijection group of degree 27 with 3 generators> gap> StructureDescription(last); "C5 x (C3 : C4)" gap> S := Semigroup([ > Bipartition([[1, 3], [2, -2], [-1], [-3]]), > Bipartition([[1, -1], [2, -2], [3], [-3]]), > Bipartition([[1], [2, -2], [3], [-1, -3]])]);; gap> GroupOfUnits(S); fail # Test IsBlockBijectionSemigroup for an ideal gap> IsBlockBijectionSemigroup(MinimalIdeal(PartitionMonoid(3))); false gap> IsBlockBijectionSemigroup(SingularDualSymmetricInverseMonoid(3)); true gap> IsBlockBijectionSemigroup(OrderEndomorphisms(3)); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `IsBlockBijectionSemigroup' on 1 argumen\ ts # Test IsPartialPermBipartitionSemigroup for an ideal gap> IsPartialPermBipartitionSemigroup(MinimalIdeal(PartitionMonoid(2))); false gap> S := AsSemigroup(IsBipartitionSemigroup, SymmetricInverseMonoid(2));; gap> IsPartialPermBipartitionSemigroup(S); true gap> IsPartialPermBipartitionSemigroup(MinimalIdeal(S)); true gap> S := Semigroup(AsSemigroup(IsBipartitionSemigroup, > SymmetricInverseMonoid(2)));; gap> IsPartialPermBipartitionSemigroup(MinimalIdeal(S)); true gap> IsPartialPermBipartitionSemigroup(OrderEndomorphisms(2)); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `IsPartialPermBipartitionSemigroup' on 1\ arguments gap> IsPermBipartitionGroup(MinimalIdeal(PartitionMonoid(2))); false gap> S := AsSemigroup(IsBipartitionSemigroup, SymmetricGroup(2));; gap> IsPermBipartitionGroup(S); true gap> IsPermBipartitionGroup(MinimalIdeal(S)); true gap> S := Semigroup(AsSemigroup(IsBipartitionSemigroup, > SymmetricInverseMonoid(2)));; gap> IsPermBipartitionGroup(MinimalIdeal(S)); false gap> IsPermBipartitionGroup(SingularDualSymmetricInverseMonoid(3)); false gap> IsPermBipartitionGroup(OrderEndomorphisms(2)); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `IsPermBipartitionGroup' on 1 arguments # Test NaturalLeqInverseSemigroup gap> NaturalLeqInverseSemigroup(DualSymmetricInverseMonoid(3)); <Operation "NaturalLeqBlockBijection"> gap> S := SymmetricInverseMonoid(3);; gap> NaturalLeqInverseSemigroup(AsSemigroup(IsBipartitionSemigroup, > S)); <Operation "NaturalLeqPartialPermBipartition"> gap> AsSemigroup(IsTransformationSemigroup, S); <transformation monoid of degree 4 with 4 generators> gap> AsSemigroup(IsBipartitionSemigroup, last); <bipartition monoid of degree 4 with 4 generators> gap> NaturalLeqInverseSemigroup(last); function( x, y ) ... end gap> NaturalLeqInverseSemigroup(PartitionMonoid(2)); Error, the argument is not an inverse semigroup # Test NaturalPartialOrder gap> NaturalPartialOrder(DualSymmetricInverseMonoid(3)); [ [ ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 9, 16 ], [ 1, 3, 8, 16 ], [ 1, 2, 11, 18 ], [ 1, 3, 11, 19 ], [ 1, 5, 8, 18 ], [ 1, 5, 9, 19 ] ] gap> o1 := NaturalPartialOrder(AsSemigroup(IsBipartitionSemigroup, > SymmetricInverseMonoid(3))); [ [ 2, 6, 7, 26, 27, 33, 34 ], [ 7, 27, 34 ], [ 4, 5, 7, 29, 30, 32, 34 ], [ 7, 30, 34 ], [ 7, 32, 34 ], [ 7, 33, 34 ], [ 34 ], [ 9, 13, 14, 23, 24, 33, 34 ], [ 14, 24, 34 ], [ 11, 12, 14, 28, 30, 31, 34 ], [ 14, 30, 34 ], [ 14, 31, 34 ], [ 14, 33, 34 ], [ 34 ], [ 16, 20, 21, 22, 24, 32, 34 ], [ 21, 24, 34 ], [ 18, 19, 21, 25, 27, 31, 34 ], [ 21, 27, 34 ], [ 21, 31, 34 ], [ 21, 32, 34 ], [ 34 ], [ 24, 32, 34 ], [ 24, 33, 34 ], [ 34 ], [ 27, 31, 34 ], [ 27, 33, 34 ], [ 34 ], [ 30, 31, 34 ], [ 30, 32, 34 ], [ 34 ], [ 34 ], [ 34 ], [ 34 ], [ ] ] gap> gr1 := DigraphReflexiveTransitiveClosure(Digraph(o1)); <immutable preorder digraph with 34 vertices, 139 edges> gap> o2 := NaturalPartialOrder(SymmetricInverseMonoid(3)); [ [ ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1 ], [ 1, 2, 6 ], [ 1, 2, 7 ], [ 1, 3, 5 ], [ 1, 3, 7 ], [ 1, 4, 5 ], [ 1, 4, 6 ], [ 1 ], [ 1 ], [ 1 ], [ 1, 5, 15 ], [ 1, 5, 16 ], [ 1, 6, 14 ], [ 1, 6, 16 ], [ 1, 7, 14 ], [ 1, 7, 15 ], [ 1, 2, 15 ], [ 1, 2, 16 ], [ 1, 2, 6, 8, 16, 20, 24 ], [ 1, 2, 7, 9, 15, 22, 23 ], [ 1, 3, 14 ], [ 1, 3, 16 ], [ 1, 3, 5, 10, 16, 18, 28 ], [ 1, 3, 7, 11, 14, 21, 27 ], [ 1, 4, 14 ], [ 1, 4, 15 ], [ 1, 4, 5, 12, 15, 17, 32 ], [ 1, 4, 6, 13, 14, 19, 31 ] ] gap> gr2 := DigraphReflexiveTransitiveClosure(Digraph(o2)); <immutable preorder digraph with 34 vertices, 139 edges> gap> IsomorphismDigraphs(gr1, gr2); (1,25,19,31,14,3,26,20,32,15,33,16,12,27,6,24,5,23,18,13,28,21,4,9,10,30,7,2, 8,29,22,17,34) # Test AsMonoid for 1 arg gap> S := Semigroup(Transformation([2, 2, 3, 4])); <commutative transformation semigroup of degree 2 with 1 generator> gap> AsMonoid(S); <trivial transformation group of degree 0 with 1 generator> gap> T := AsSemigroup(IsBipartitionSemigroup, S);; gap> IsMonoid(T); false gap> IsMonoidAsSemigroup(T); true gap> AsMonoid(T); <trivial block bijection group of degree 1 with 1 generator> gap> S := Semigroup(Bipartition([[1, 2, -1, -3], [3, -2]]), > Bipartition([[1, 2, -2, -3], [3, -1]]), > Bipartition([[1, 2, -3], [3, -1, -2]]), > Bipartition([[1, 3, -1, -2], [2, -3]]), > Bipartition([[1, -3], [2, 3, -1, -2]])); <block bijection semigroup of degree 3 with 5 generators> gap> S := AsMonoid(S); fail # Test GeneratorsOfInverseSemigroup gap> S := Monoid(Bipartition([[1, -2], [2, -3], [3, -1]]), > Bipartition([[1, -2], [2, -1], [3, -3]]), > Bipartition([[1], [2, -1], [3, -2], [-3]]), > Bipartition([[1, -2], [2, -3], [3], [-1]]));; gap> IsInverseSemigroup(S); true gap> HasGeneratorsOfInverseSemigroup(S); false gap> GeneratorsOfInverseSemigroup(S); [ <block bijection: [ 1, -1 ], [ 2, -2 ], [ 3, -3 ]>, <block bijection: [ 1, -2 ], [ 2, -3 ], [ 3, -1 ]>, <block bijection: [ 1, -2 ], [ 2, -1 ], [ 3, -3 ]>, <bipartition: [ 1 ], [ 2, -1 ], [ 3, -2 ], [ -3 ]> ] gap> InverseSemigroup(last) = S; true gap> GeneratorsOfInverseMonoid(S); [ <block bijection: [ 1, -2 ], [ 2, -3 ], [ 3, -1 ]>, <block bijection: [ 1, -2 ], [ 2, -1 ], [ 3, -3 ]>, <bipartition: [ 1 ], [ 2, -1 ], [ 3, -2 ], [ -3 ]> ] gap> InverseMonoid(last) = S; true # Operator < gap> A := Semigroup(Bipartition([[1, 2], [-1], [-2]]), > Bipartition([[1, 2, -1, -2]])); <bipartition semigroup of degree 2 with 2 generators> gap> B := Semigroup([Bipartition([[1, 2, -2], [-1]]), > Bipartition([[1, 2, -1], [-2]])]); <bipartition semigroup of degree 2 with 2 generators> gap> C := Monoid([Bipartition([[1, -1]])]); <trivial block bijection group of degree 1 with 1 generator> gap> A < B; true gap> B < A; false gap> B = A; false gap> C = A; false gap> C < A; true gap> C < B; true # gap> SEMIGROUPS.StopTest(); gap> STOP_TEST("Semigroups package: standard/semigroups/semibipart.tst"); [Dauer der Verarbeitung: 0.49 Sekunden, vorverarbeitet 2026-05-01] |
2026-05-26