| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/semigroups/tst/standard/semigroups/ (Algebra von RWTH Aachen Version 4.15.1©) Datei vom 29.7.2025 mit Größe 101 kB |
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Quelle semirms.tst Sprache: unbekannt |
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Spracherkennung für: .tst vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen] ############################################################################
## #W standard/semigroups/semirms.tst #Y Copyright (C) 2015-2022 James D. Mitchell ## Wilf A. Wilson ## Finn Smith ## ## Licensing information can be found in the README file of this package. ## ############################################################################# ## #@local BruteForceInverseCheck, BruteForceIsoCheck, F, G, H, I, R, RR, S, T, U #@local a, an, b, c, comps, d, data, e, filename, first_occurrence, func, ht, i #@local id, idems, inv, iso, map, mat, rels, x, y, z, zero gap> START_TEST("Semigroups package: standard/semigroups/semirms.tst"); gap> LoadPackage("semigroups", false);; # gap> SEMIGROUPS.StartTest();; # helper functions gap> BruteForceIsoCheck := function(iso) > local x, y; > if not IsInjective(iso) or not IsSurjective(iso) then > return false; > fi; > for x in Source(iso) do > for y in 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;; # AsSemigroup: # convert from IsPBRSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > PBR([[-1], [-5], [-1], [-1], [-5], [-5], [-1]], > [[1, 3, 4, 7], [], [], [], [2, 5, 6], [], []]), > PBR([[-4], [-2], [-4], [-4], [-2], [-2], [-2]], > [[], [2, 5, 6, 7], [], [1, 3, 4], [], [], []]), > PBR([[-3], [-6], [-3], [-3], [-6], [-6], [-3]], > [[], [], [1, 3, 4, 7], [], [], [2, 5, 6], []])]); <pbr semigroup of degree 7 with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpSemigroup to IsReesMatrixSemigroup gap> F := FreeSemigroup(3);; AssignGeneratorVariables(F);; gap> rels := [[s1 ^ 2, s1], > [s1 * s3, s3], > [s2 ^ 2, s2], > [s3 * s1, s1], > [s3 * s2, s1 * s2], > [s3 ^ 2, s3], > [s1 * s2 * s1, s1], > [s1 * s2 * s3, s3], > [s2 * s1 * s2, s2]];; gap> S := F / rels; <fp semigroup with 3 generators and 9 relations of length 34> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Bipartition([[1, 3, 4, 7, -1], [2, 5, 6, -5], [-2], [-3], [-4], [-6], [-7]]), > Bipartition([[1, 3, 4, -4], [2, 5, 6, 7, -2], [-1], [-3], [-5], [-6], [-7]]), > Bipartition([[1, 3, 4, 7, -3], [2, 5, 6, -6], [-1], [-2], [-4], [-5], [-7]])]); <bipartition semigroup of degree 7 with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Transformation([1, 5, 1, 1, 5, 5, 1]), > Transformation([4, 2, 4, 4, 2, 2, 2]), > Transformation([3, 6, 3, 3, 6, 6, 3])]); <transformation semigroup of degree 7 with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsBooleanMat, > [[true, false, false, false, false, false, false], > [false, false, false, false, true, false, false], > [true, false, false, false, false, false, false], > [true, false, false, false, false, false, false], > [false, false, false, false, true, false, false], > [false, false, false, false, true, false, false], > [true, false, false, false, false, false, false]]), > Matrix(IsBooleanMat, > [[false, false, false, true, false, false, false], > [false, true, false, false, false, false, false], > [false, false, false, true, false, false, false], > [false, false, false, true, false, false, false], > [false, true, false, false, false, false, false], > [false, true, false, false, false, false, false], > [false, true, false, false, false, false, false]]), > Matrix(IsBooleanMat, > [[false, false, true, false, false, false, false], > [false, false, false, false, false, true, false], > [false, false, true, false, false, false, false], > [false, false, true, false, false, false, false], > [false, false, false, false, false, true, false], > [false, false, false, false, false, true, false], > [false, false, true, false, false, false, false]])]); <semigroup of 7x7 boolean matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity]])]); <semigroup of 7x7 max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsMinPlusMatrix, > [[0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity]])]); <semigroup of 7x7 min-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsProjectiveMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity]])]); <semigroup of 7x7 projective max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(Integers, > [[1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [1, 0, 0, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [1, 0, 0, 0, 0, 0, 0]]), > Matrix(Integers, > [[0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0]]), > Matrix(Integers, > [[0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0, 0]])]); <semigroup of 7x7 integer matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity, -infinity]], > 1), > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity, -infinity]], > 1), > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity, -infinity]], > 1)]); <semigroup of 7x7 tropical max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMinPlusMatrix, > [[0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity, infinity]], 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity, infinity]], 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity, infinity]], > 3)]); <semigroup of 7x7 tropical min-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixSemigroup to IsReesMatrixSemigroup gap> S := Semigroup([ > Matrix(IsNTPMatrix, > [[1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [1, 0, 0, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0, 0], > [1, 0, 0, 0, 0, 0, 0]], 4, 1), > Matrix(IsNTPMatrix, > [[0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0], > [0, 0, 0, 1, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0, 0]], 4, 1), > Matrix(IsNTPMatrix, > [[0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0, 0]], 4, 1)]); <semigroup of 7x7 ntp matrices with 3 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x3 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPBRSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > PBR([[-4], [-5], [-3], [-1], [-2], [-1]], > [[4, 6], [5], [3], [1], [2], []]), > PBR([[-1], [-2], [-3], [-4], [-5], [-2]], > [[1], [2, 6], [3], [4], [5], []]), > PBR([[-3], [-3], [-3], [-3], [-3], [-3]], > [[], [], [1, 2, 3, 4, 5, 6], [], [], []])]); <pbr semigroup of degree 6 with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpSemigroup to IsReesZeroMatrixSemigroup gap> F := FreeSemigroup(3);; AssignGeneratorVariables(F);; gap> rels := [[s1 * s2, s1], > [s1 * s3, s3], > [s2 ^ 2, s2], > [s2 * s3, s3], > [s3 * s1, s3], > [s3 * s2, s3], > [s3 ^ 2, s3], > [s1 ^ 3, s1], > [s2 * s1 ^ 2, s2]];; gap> S := F / rels; <fp semigroup with 3 generators and 9 relations of length 32> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Bipartition([[1, -4], [2, -5], [3, -3], [4, 6, -1], [5, -2], [-6]]), > Bipartition([[1, -1], [2, 6, -2], [3, -3], [4, -4], [5, -5], [-6]]), > Bipartition([[1, 2, 3, 4, 5, 6, -3], [-1], [-2], [-4], [-5], [-6]])]); <bipartition semigroup of degree 6 with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S);; gap> (IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group((1, 4)(2, 5))) > or (not IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group((1, 2))); true gap> Length(Rows(T)) = 2 and Length(Columns(T)) = 1; 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Transformation([4, 5, 3, 1, 2, 1]), > Transformation([1, 2, 3, 4, 5, 2]), > Transformation([3, 3, 3, 3, 3, 3])]); <transformation semigroup of degree 6 with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S);; gap> (IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group((1, 4)(2, 5))) > or (not IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group((1, 2))); true gap> Length(Rows(T)) = 2 and Length(Columns(T)) = 1; 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsBooleanMat, > [[false, false, false, true, false, false], > [false, false, false, false, true, false], > [false, false, true, false, false, false], > [true, false, false, false, false, false], > [false, true, false, false, false, false], > [true, false, false, false, false, false]]), > Matrix(IsBooleanMat, > [[true, false, false, false, false, false], > [false, true, false, false, false, false], > [false, false, true, false, false, false], > [false, false, false, true, false, false], > [false, false, false, false, true, false], > [false, true, false, false, false, false]]), > Matrix(IsBooleanMat, > [[false, false, true, false, false, false], > [false, false, true, false, false, false], > [false, false, true, false, false, false], > [false, false, true, false, false, false], > [false, false, true, false, false, false], > [false, false, true, false, false, false]])]); <semigroup of 6x6 boolean matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity]])]); <semigroup of 6x6 max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity], > [infinity, 0, infinity, infinity, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity]])]); <semigroup of 6x6 min-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity]])]); <semigroup of 6x6 projective max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(Integers, > [[0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0], > [1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0]]), > Matrix(Integers, > [[1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0], > [0, 1, 0, 0, 0, 0]]), > Matrix(Integers, > [[0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0]])]); <semigroup of 6x6 integer matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [0, -infinity, -infinity, -infinity, -infinity, -infinity]], 1), > Matrix(IsTropicalMaxPlusMatrix, > [[0, -infinity, -infinity, -infinity, -infinity, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, -infinity, 0, -infinity, -infinity], > [-infinity, -infinity, -infinity, -infinity, 0, -infinity], > [-infinity, 0, -infinity, -infinity, -infinity, -infinity]], 1), > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity], > [-infinity, -infinity, 0, -infinity, -infinity, -infinity]], 1)]); <semigroup of 6x6 tropical max-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity], > [0, infinity, infinity, infinity, infinity, infinity]], 3), > Matrix(IsTropicalMinPlusMatrix, > [[0, infinity, infinity, infinity, infinity, infinity], > [infinity, 0, infinity, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, infinity, 0, infinity, infinity], > [infinity, infinity, infinity, infinity, 0, infinity], > [infinity, 0, infinity, infinity, infinity, infinity]], 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity], > [infinity, infinity, 0, infinity, infinity, infinity]], 3)]); <semigroup of 6x6 tropical min-plus matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := Semigroup([ > Matrix(IsNTPMatrix, > [[0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0], > [0, 0, 1, 0, 0, 0], > [1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0], > [1, 0, 0, 0, 0, 0]], 4, 1), > Matrix(IsNTPMatrix, > [[1, 0, 0, 0, 0, 0], > [0, 1, 0, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 0, 1, 0, 0], > [0, 0, 0, 0, 1, 0], > [0, 1, 0, 0, 0, 0]], 4, 1), > Matrix(IsNTPMatrix, > [[0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0], > [0, 0, 1, 0, 0, 0]], 4, 1)]); <semigroup of 6x6 ntp matrices with 3 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPBRMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > PBR([[-3], [-2], [-1]], [[3], [2], [1]]), > PBR([[-2], [-2], [-2]], [[], [1, 2, 3], []])]); <pbr monoid of degree 3 with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpMonoid to IsReesZeroMatrixSemigroup gap> F := FreeMonoid(2);; AssignGeneratorVariables(F);; gap> rels := [[m1 ^ 2, One(F)], > [m1 * m2, m2], > [m2 * m1, m2], > [m2 ^ 2, m2]];; gap> S := F / rels; <fp monoid with 2 generators and 4 relations of length 13> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Bipartition([[1, -3], [2, -2], [3, -1]]), > Bipartition([[1, 2, 3, -2], [-1], [-3]])]); <bipartition monoid of degree 3 with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S);; gap> (IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group([(1, 3)])) > or (not IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group([(1, 2)])); true gap> Length(Rows(T)) = 1 and Length(Columns(T)) = 1; 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Transformation([3, 2, 1]), Transformation([2, 2, 2])]); <transformation monoid of degree 3 with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S);; gap> (IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group([(1, 3)])) > or (not IsActingSemigroup(S) and UnderlyingSemigroup(T) = Group([(1, 2)])); true gap> Length(Rows(T)) = 1 and Length(Columns(T)) = 1; 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsBooleanMat, > [[false, false, true], [false, true, false], > [true, false, false]]), > Matrix(IsBooleanMat, > [[false, true, false], > [false, true, false], > [false, true, false]])]); <monoid of 3x3 boolean matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsMaxPlusMatrix, > [[-infinity, -infinity, 0], > [-infinity, 0, -infinity], > [0, -infinity, -infinity]]), > Matrix(IsMaxPlusMatrix, > [[-infinity, 0, -infinity], > [-infinity, 0, -infinity], > [-infinity, 0, -infinity]])]); <monoid of 3x3 max-plus matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsMinPlusMatrix, > [[infinity, infinity, 0], > [infinity, 0, infinity], > [0, infinity, infinity]]), > Matrix(IsMinPlusMatrix, > [[infinity, 0, infinity], > [infinity, 0, infinity], > [infinity, 0, infinity]])]); <monoid of 3x3 min-plus matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, -infinity, 0], > [-infinity, 0, -infinity], > [0, -infinity, -infinity]]), > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, 0, -infinity], > [-infinity, 0, -infinity], > [-infinity, 0, -infinity]])]); <monoid of 3x3 projective max-plus matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(Integers, > [[0, 0, 1], > [0, 1, 0], > [1, 0, 0]]), > Matrix(Integers, > [[0, 1, 0], > [0, 1, 0], > [0, 1, 0]])]); <monoid of 3x3 integer matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, -infinity, 0], > [-infinity, 0, -infinity], > [0, -infinity, -infinity]], 1), > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, 0, -infinity], > [-infinity, 0, -infinity], > [-infinity, 0, -infinity]], 1)]); <monoid of 3x3 tropical max-plus matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, infinity, 0], > [infinity, 0, infinity], > [0, infinity, infinity]], 3), > Matrix(IsTropicalMinPlusMatrix, > [[infinity, 0, infinity], > [infinity, 0, infinity], > [infinity, 0, infinity]], 3)]); <monoid of 3x3 tropical min-plus matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixMonoid to IsReesZeroMatrixSemigroup gap> S := Monoid([ > Matrix(IsNTPMatrix, > [[0, 0, 1], > [0, 1, 0], > [1, 0, 0]], 4, 1), > Matrix(IsNTPMatrix, > [[0, 1, 0], > [0, 1, 0], > [0, 1, 0]], 4, 1)]); <monoid of 3x3 ntp matrices with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpMonoid to IsReesZeroMatrixSemigroup gap> F := FreeMonoid(1);; AssignGeneratorVariables(F);; gap> rels := [[m1 ^ 2, m1]];; gap> S := F / rels; <fp monoid with 1 generator and 1 relation of length 4> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group(())> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPBRMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > PBR([[-2], [-1]], [[2], [1]])]); <commutative pbr monoid of degree 2 with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpMonoid to IsReesMatrixSemigroup gap> F := FreeMonoid(1);; AssignGeneratorVariables(F);; gap> rels := [[m1 ^ 2, One(F)]];; gap> S := F / rels; <fp monoid with 1 generator and 1 relation of length 3> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionMonoid to IsReesMatrixSemigroup gap> S := InverseMonoid([ > Bipartition([[1, -1], [2, -2]]), > Bipartition([[1, -2], [2, -1]])]); <block bijection group of degree 2 with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionSemigroup to IsReesMatrixSemigroup gap> S := InverseSemigroup([ > Bipartition([[1, -1], [2, -2]]), > Bipartition([[1, -2], [2, -1]])]); <block bijection group of degree 2 with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionMonoid to IsReesZeroMatrixSemigroup gap> S := InverseMonoid(Bipartition([[1, -1], [2, -2]]), > Bipartition([[1, -2], [2, -1]]), > Bipartition([[1, 2, -1, -2]])); <inverse block bijection monoid of degree 2 with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBipartitionSemigroup to IsReesZeroMatrixSemigroup gap> S := InverseSemigroup(Bipartition([[1, -1], [2, -2]]), > Bipartition([[1, -2], [2, -1]]), > Bipartition([[1, 2, -1, -2]])); <inverse block bijection monoid of degree 2 with 2 generators> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermSemigroup to IsReesMatrixSemigroup gap> S := InverseSemigroup(PartialPerm([1, 2], [2, 1]));; gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermMonoid to IsReesMatrixSemigroup gap> S := InverseMonoid(PartialPerm([1, 2], [2, 1]));; gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermSemigroup to IsReesZeroMatrixSemigroup gap> S := InverseSemigroup(PartialPerm([1, 2], [2, 1]), > PartialPerm([]));; gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPartialPermMonoid to IsReesZeroMatrixSemigroup gap> S := InverseMonoid(PartialPerm([1, 2], [2, 1]), > PartialPerm([]));; gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTransformationMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Transformation([2, 1])]); <commutative transformation monoid of degree 2 with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsBooleanMatMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsBooleanMat, > [[false, true], [true, false]])]); <commutative monoid of 2x2 boolean matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMaxPlusMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsMaxPlusMatrix, > [[-infinity, 0], [0, -infinity]])]); <commutative monoid of 2x2 max-plus matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsMinPlusMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsMinPlusMatrix, > [[infinity, 0], [0, infinity]])]); <commutative monoid of 2x2 min-plus matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsProjectiveMaxPlusMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsProjectiveMaxPlusMatrix, > [[-infinity, 0], [0, -infinity]])]); <commutative monoid of 2x2 projective max-plus matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsIntegerMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(Integers, > [[0, 1], [1, 0]])]); <commutative monoid of 2x2 integer matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMaxPlusMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsTropicalMaxPlusMatrix, > [[-infinity, 0], [0, -infinity]], 1)]); <commutative monoid of 2x2 tropical max-plus matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsTropicalMinPlusMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsTropicalMinPlusMatrix, > [[infinity, 0], [0, infinity]], 3)]); <commutative monoid of 2x2 tropical min-plus matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsNTPMatrixMonoid to IsReesMatrixSemigroup gap> S := Monoid([ > Matrix(IsNTPMatrix, > [[0, 1], [1, 0]], 4, 1)]); <commutative monoid of 2x2 ntp matrices with 1 generator> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsFpMonoid to IsReesMatrixSemigroup gap> F := FreeMonoid(0);; AssignGeneratorVariables(F);; gap> rels := [];; gap> S := F / rels;; gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsPermGroup to IsReesMatrixSemigroup gap> S := DihedralGroup(IsPermGroup, 4); Group([ (1,2), (3,4) ]) gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,3)(2,4), (1,4)(2,3) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsGroup to IsReesMatrixSemigroup gap> S := DihedralGroup(4); <pc group of size 4 with 2 generators> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group([ (1,2)(3,4), (1,3)(2,4) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesMatrixSemigroup to IsReesMatrixSemigroup gap> S := ReesMatrixSemigroup(Group([(1, 2)]), [[(1, 2), (1, 2)], [(), ()]]); <Rees matrix semigroup 2x2 over Group([ (1,2) ])> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 2x2 over Group([ (1,2) ])> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsReesZeroMatrixSemigroup to IsReesZeroMatrixSemigroup gap> S := ReesZeroMatrixSemigroup(Group([(1, 2)]), > [[(1, 2), (1, 2)], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> T := AsSemigroup(IsReesZeroMatrixSemigroup, S); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> 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(IsReesZeroMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from IsGraphInverseSemigroup to IsReesMatrixSemigroup gap> S := GraphInverseSemigroup(Digraph([[]])); <finite graph inverse semigroup with 1 vertex, 0 edges> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # AsSemigroup: # convert from a free band to IsReesMatrixSemigroup gap> S := FreeBand(1); <free band on the generators [ x1 ]> gap> T := AsSemigroup(IsReesMatrixSemigroup, S); <Rees matrix semigroup 1x1 over Group(())> 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(IsReesMatrixSemigroup, S);; gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # RMSNormalization 1: # for a Rees matrix semigroup over a group without an inverse op gap> G := Semigroup([ > Transformation([4, 4, 2, 3, 4]), > Transformation([4, 4, 3, 2, 4])]); <transformation semigroup of degree 5 with 2 generators> gap> IsGroup(G); false gap> IsGroupAsSemigroup(G); true gap> mat := [ > [Transformation([4, 4, 2, 3, 4]), Transformation([4, 4, 3, 2, 4])], > [Transformation([2, 2, 4, 3, 2]), Transformation([3, 3, 4, 2, 3])]];; gap> R := ReesMatrixSemigroup(G, mat);; gap> iso := RMSNormalization(R);; gap> inv := InverseGeneralMapping(iso);; gap> ForAll(R, x -> (x ^ iso) ^ inv = x); true # RMSNormalization 2: # for a Rees matrix semigroup over IsGroup gap> R := ReesMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), (1, 3), (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]); <Rees matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )> gap> iso := RMSNormalization(R);; gap> Matrix(Range(iso)); [ [ (), (), () ], [ (), (1,2,4), (1,4) ], [ (), (1,2)(3,4), (1,4)(2,3) ], [ (), (1,3)(2,4), (1,4,3,2) ] ] # RMSNormalization 3: # error checking gap> G := FullTransformationMonoid(4);; gap> RMSNormalization(ReesMatrixSemigroup(G, [[IdentityTransformation]])); Error, the underlying semigroup of the argument (a subsemigroup of a Rees mat\ rix semigroup) does not satisfy IsGroupAsSemigroup # RZMSNormalization 1: # for a Rees 0-matrix semigroup over a group without an inverse op gap> G := Semigroup([ > Transformation([1, 3, 2, 1]), > Transformation([3, 1, 2, 3])]); <transformation semigroup of degree 4 with 2 generators> gap> IsGroup(G); false gap> IsGroupAsSemigroup(G); true gap> mat := [ > [Transformation([2, 1, 3, 2]), 0, 0], > [0, 0, Transformation([2, 3, 1, 2])], > [0, Transformation([3, 1, 2, 3]), Transformation([2, 3, 1, 2])]];; gap> R := ReesZeroMatrixSemigroup(G, mat);; gap> iso := RZMSNormalization(R);; gap> inv := InverseGeneralMapping(iso);; gap> ForAll(R, x -> (x ^ iso) ^ inv = x); true # RZMSNormalization 2: # for a Rees 0-matrix semigroup over the symmetric group S4 gap> G := SymmetricGroup(4);; gap> R := ReesZeroMatrixSemigroup(G, > [[0, (1, 4)(2, 3), 0], [0, 0, (4, 2, 3)], [(2, 4, 3), 0, 0]]); <Rees 0-matrix semigroup 3x3 over Sym( [ 1 .. 4 ] )> gap> IsInverseSemigroup(R); true gap> iso := RZMSNormalization(R); <Rees 0-matrix semigroup 3x3 over Sym( [ 1 .. 4 ] )> -> <Rees 0-matrix semigroup 3x3 over Sym( [ 1 .. 4 ] )> gap> S := Range(iso); <Rees 0-matrix semigroup 3x3 over Sym( [ 1 .. 4 ] )> gap> Matrix(S); [ [ (), 0, 0 ], [ 0, (), 0 ], [ 0, 0, () ] ] gap> inv := InverseGeneralMapping(iso);; gap> x := MultiplicativeZero(R) ^ iso; 0 gap> x ^ inv = MultiplicativeZero(R); true gap> x := RMSElement(R, 1, (), 1); (1,(),1) gap> x ^ iso; (1,(2,4,3),2) gap> (x ^ iso) ^ inv = x; true gap> G := SymmetricGroup(4);; gap> mat := [ > [0, 0, (1, 3, 2), 0, (), 0, 0, (1, 2, 3)], > [(), 0, 0, 0, 0, (1, 3, 4, 2), 0, (2, 4)], > [0, 0, 0, (1, 2, 3), 0, 0, (1, 3, 2), 0], > [0, 0, 0, 0, 0, 0, (1, 4, 2, 3), 0], > [(), (1, 2, 3), (1, 2), 0, 0, 0, 0, 0], > [0, (), 0, 0, 0, (1, 2), 0, 0]];; gap> R := ReesZeroMatrixSemigroup(G, mat); <Rees 0-matrix semigroup 8x6 over Sym( [ 1 .. 4 ] )> gap> iso := RZMSNormalization(R); <Rees 0-matrix semigroup 8x6 over Sym( [ 1 .. 4 ] )> -> <Rees 0-matrix semigroup 8x6 over Sym( [ 1 .. 4 ] )> gap> S := Range(iso); <Rees 0-matrix semigroup 8x6 over Sym( [ 1 .. 4 ] )> # check that mat is in the 'normal' form gap> mat := Matrix(S); [ [ (), (), (), 0, 0, 0, 0, 0 ], [ (), 0, 0, (), (), 0, 0, 0 ], [ 0, 0, (), (1,4,2), 0, (), 0, 0 ], [ 0, 0, 0, 0, (), (2,3,4), 0, 0 ], [ 0, 0, 0, 0, 0, 0, (), () ], [ 0, 0, 0, 0, 0, 0, 0, () ] ] gap> first_occurrence := l -> First([1 .. Length(l)], i -> l[i] <> 0);; gap> x := Length(mat);; gap> ForAll([1 .. x - 1], > i -> first_occurrence(mat[i]) <= first_occurrence(mat[i + 1])); true gap> ForAll([1 .. Length(mat[1]) - 1], i -> > first_occurrence(mat{[1 .. x]}[i]) <= first_occurrence(mat{[1 .. x]}[i + 1])); true # check that the connected components are grouped together gap> comps := RZMSConnectedComponents(S); [ [ [ 1, 2, 3, 4, 5, 6 ], [ 1, 2, 3, 4 ] ], [ [ 7, 8 ], [ 5, 6 ] ] ] gap> Concatenation(List(comps, x -> x[1])) = Rows(R); true gap> Concatenation(List(comps, x -> x[2])) = Columns(R); true # RZMSNormalization 3: # for a Rees 0-matrix semigroup over a non-IsGroup group, # and a non-group semigroup gap> T := FullTransformationMonoid(4);; gap> G := GroupOfUnits(T);; gap> R := ReesZeroMatrixSemigroup(T, > [[0, IdentityTransformation], [IdentityTransformation, 0]]); <Rees 0-matrix semigroup 2x2 over <full transformation monoid of degree 4>> gap> RZMSNormalization(R);; gap> Matrix(Range(last)); [ [ IdentityTransformation, 0 ], [ 0, IdentityTransformation ] ] gap> R := ReesZeroMatrixSemigroup(G, [[Transformation([3, 1, 4, 2])]]);; gap> RZMSNormalization(R);; gap> Matrix(Range(last)); [ [ IdentityTransformation ] ] # RZMSNormalization 4: # for a Rees 0-matrix semigroup with some all-zero rows/columns gap> G := Group(());; gap> R := ReesZeroMatrixSemigroup(G, [[0, 0], [0, ()]]);; gap> Matrix(Range(RZMSNormalization(R))); [ [ (), 0 ], [ 0, 0 ] ] gap> R := ReesZeroMatrixSemigroup(G, [[0, 0], [(), ()]]);; gap> Matrix(Range(RZMSNormalization(R))); [ [ (), () ], [ 0, 0 ] ] gap> R := ReesZeroMatrixSemigroup(G, [[0, ()], [0, ()]]);; gap> Matrix(Range(RZMSNormalization(R))); [ [ (), 0 ], [ (), 0 ] ] # IsInverseSemigroup for a Rees 0-matrix semigroup 1: # easy true examples gap> R := ReesZeroMatrixSemigroup(Group(()), [[()]]); <Rees 0-matrix semigroup 1x1 over Group(())> gap> IsInverseSemigroup(R); true gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); true # gap> T := Semigroup(Transformation([2, 1])); <commutative transformation semigroup of degree 2 with 1 generator> gap> IsGroupAsSemigroup(T); true gap> R := ReesZeroMatrixSemigroup(T, [[Transformation([2, 1])]]);; gap> IsInverseSemigroup(R); true gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); true # IsInverseSemigroup for a Rees 0-matrix semigroup 2: # false because of underlying semigroup gap> x := Transformation([1, 1, 2]);; gap> T := Semigroup(x);; gap> IsInverseSemigroup(T); false gap> R := ReesZeroMatrixSemigroup(T, [[0, x], [0, x ^ 2]]);; gap> IsInverseSemigroup(R); false gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); false # T is known not to be regular gap> T := Semigroup(x);; gap> IsRegularSemigroup(T); false gap> R := ReesZeroMatrixSemigroup(T, [[0, x], [0, x ^ 2]]);; gap> IsInverseSemigroup(R); false gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); false # T is known not to be a monoid gap> T := Semigroup(x);; gap> IsMonoidAsSemigroup(T); false gap> R := ReesZeroMatrixSemigroup(T, [[0, x], [0, x ^ 2]]);; gap> IsInverseSemigroup(R); false gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); false # T is known not to have group of units gap> T := Semigroup(x);; gap> GroupOfUnits(T); fail gap> R := ReesZeroMatrixSemigroup(T, [[0, x], [0, x ^ 2]]);; gap> IsInverseSemigroup(R); false gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); false # T does not have a group of units gap> T := Semigroup(x);; gap> R := ReesZeroMatrixSemigroup(T, [[x, 0], [0, x ^ 2]]);; gap> IsInverseSemigroup(R); false gap> IsInverseSemigroup(AsSemigroup(IsTransformationSemigroup, R)); false # IsInverseSemigroup for a Rees 0-matrix semigroup 3: # false because of matrix gap> S := Semigroup(SymmetricInverseMonoid(3)); <partial perm monoid of rank 3 with 4 generators> gap> id := Identity(S); <identity partial perm on [ 1, 2, 3 ]> gap> zero := MultiplicativeZero(S); <empty partial perm> # Non-diagonal matrix: Rows or columns without precisely one non-zero entry gap> R := ReesZeroMatrixSemigroup(S, [[0, id, 0], [id, 0, 0], [0, 0, 0]]);; gap> IsInverseSemigroup(R); false gap> R := ReesZeroMatrixSemigroup(S, [[0, 0, 0], [id, 0, 0], [0, id, 0]]);; gap> IsInverseSemigroup(R); false gap> R := ReesZeroMatrixSemigroup(S, [[0, 0, id], [id, id, 0], [0, id, 0]]);; gap> IsInverseSemigroup(R); false gap> R := ReesZeroMatrixSemigroup(S, [[0, id, 0], [0, id, 0], [0, id, 0]]);; gap> IsInverseSemigroup(R); false gap> R := ReesZeroMatrixSemigroup(S, [[id, 0, 0], [id, id, 0], [0, id, 0]]);; gap> IsInverseSemigroup(R); false gap> R := ReesZeroMatrixSemigroup(S, [[zero, id]]);; gap> IsInverseSemigroup(R); # Non-square matrix false gap> R := ReesZeroMatrixSemigroup(S, [[id, 0, 0], [0, 0, id], [0, zero, 0]]);; gap> IsInverseSemigroup(R); # Matrix entries not in the group of units false gap> y := PartialPerm([1, 2, 0]); <identity partial perm on [ 1, 2 ]> gap> R := ReesZeroMatrixSemigroup(S, [[id, 0, 0], [0, 0, id], [0, y, 0]]);; gap> IsInverseSemigroup(R); false gap> T := FullTransformationMonoid(3);; gap> R := ReesZeroMatrixSemigroup(T, [[Identity(T)]]);; gap> IsInverseSemigroup(R); # Semigroup is not an inverse monoid false gap> y := PartialPerm([3, 2, 1]);; gap> R := ReesZeroMatrixSemigroup(S, [[id, 0, 0], [0, id, 0], [0, 0, y]]);; gap> IsInverseSemigroup(R); true # NrIdempotents and Idempotents 1: # for an inverse Rees 0-matrix semigroup gap> S := SymmetricInverseMonoid(4); <symmetric inverse monoid of degree 4> gap> x := PartialPerm([2, 1, 4, 3]);; gap> y := PartialPerm([2, 4, 3, 1]);; gap> R := ReesZeroMatrixSemigroup(S, [[0, x], [y, 0]]); <Rees 0-matrix semigroup 2x2 over <symmetric inverse monoid of degree 4>> gap> IsInverseSemigroup(R); true gap> NrIdempotents(R); 33 gap> NrIdempotents(R) = NrIdempotents(S) * Length(Rows(R)) + 1; true gap> idems := Idempotents(R);; gap> IsDuplicateFreeList(idems); true gap> Length(idems) = NrIdempotents(R); true gap> ForAll(idems, IsIdempotent); true # NrIdempotents and Idempotents 2: # for an inverse Rees 0-matrix semigroup over a group gap> R := ReesZeroMatrixSemigroup(Group(()), [[()]]); <Rees 0-matrix semigroup 1x1 over Group(())> gap> NrIdempotents(R); 2 gap> Idempotents(R); [ 0, (1,(),1) ] gap> AsSet(Idempotents(R)) = Elements(R); true gap> IsBand(R); true gap> x := Transformation([2, 1]);; gap> T := Semigroup(x); <commutative transformation semigroup of degree 2 with 1 generator> gap> R := ReesZeroMatrixSemigroup(T, [[x, 0], [x, x ^ 2]]);; gap> NrIdempotents(R); 4 gap> Idempotents(R); [ 0, (1,Transformation( [ 2, 1 ] ),1), (1,Transformation( [ 2, 1 ] ),2), (2,IdentityTransformation,2) ] gap> ForAll(Idempotents(R), IsIdempotent); true gap> x := Transformation([1, 1, 2]);; gap> T := Semigroup(x); <commutative transformation semigroup of degree 3 with 1 generator> gap> R := ReesZeroMatrixSemigroup(T, [[x, 0], [0, x ^ 2]]);; gap> NrIdempotents(R); 3 gap> i := ShallowCopy(Idempotents(R));; gap> Sort(i); gap> i; [ 0, (1,Transformation( [ 1, 1, 1 ] ),1), (2,Transformation( [ 1, 1, 1 ] ),2) ] gap> ForAll(Idempotents(R), IsIdempotent); true # NrIdempotents and Idempotents 3: # for an sub-RZMS of an Rees 0-matrix semigroup gap> S := SymmetricInverseMonoid(4);; gap> x := PartialPerm([2, 1, 4, 3]);; gap> y := PartialPerm([2, 4, 3, 1]);; gap> z := PartialPerm([0, 0, 0, 0]);; gap> R := ReesZeroMatrixSemigroup(S, [[x, x, 0], [y, 0, 0], [0, 0, x]]);; gap> IsInverseSemigroup(R); false # gap> T := Semigroup(RMSElement(R, 1, x, 1));; gap> IsReesZeroMatrixSemigroup(T); false gap> NrIdempotents(T); 1 gap> Idempotents(T); [ (1,(1,2)(3,4),1) ] gap> T := Semigroup(RMSElement(R, 1, x, 1));; gap> IsReesZeroMatrixSemigroup(T); false gap> NrIdempotents(T); 1 gap> Idempotents(T); [ (1,(1,2)(3,4),1) ] gap> T := Semigroup(RMSElement(R, 1, y ^ -1, 2));; gap> IsInverseSemigroup(T); true gap> NrIdempotents(T); 1 gap> T := Semigroup(RMSElement(R, 1, y ^ -1, 2));; gap> IsInverseSemigroup(T); true gap> Idempotents(T); [ (1,(1,4,2)(3),2) ] gap> T := Semigroup(RMSElement(R, 1, y ^ -1, 2));; gap> NrIdempotents(T); 1 gap> T := Semigroup(RMSElement(R, 1, y ^ -1, 2));; gap> Idempotents(T); [ (1,(1,4,2)(3),2) ] gap> T := Semigroup(RMSElement(R, 1, y ^ -1, 2));; gap> SetIsInverseSemigroup(T, true); gap> Idempotents(T); [ (1,(1,4,2)(3),2) ] # gap> S := SymmetricInverseMonoid(4);; gap> x := PartialPerm([2, 1, 4, 3]);; gap> y := PartialPerm([2, 4, 3, 1]);; gap> z := PartialPerm([0, 0, 0, 0]);; gap> R := ReesZeroMatrixSemigroup(S, [[x, x, 0], [y, 0, 0], [0, 0, x]]);; gap> IsInverseSemigroup(R); false gap> T := ReesZeroMatrixSubsemigroup(R, [2, 3], S, [1, 2, 3]); <Rees 0-matrix semigroup 2x3 over <symmetric inverse monoid of degree 4>> gap> IsInverseSemigroup(T); false gap> T := ReesZeroMatrixSubsemigroup(R, [2, 3], S, [1, 2]); <Rees 0-matrix semigroup 2x2 over <symmetric inverse monoid of degree 4>> gap> IsInverseSemigroup(T); false gap> T := ReesZeroMatrixSubsemigroup(R, [1, 2], S, [2, 3]); <Rees 0-matrix semigroup 2x2 over <symmetric inverse monoid of degree 4>> gap> IsInverseSemigroup(T); false gap> T := ReesZeroMatrixSubsemigroup(R, [2, 3], S, [1, 3]);; gap> IsInverseSemigroup(T) and NrIdempotents(T) = 33; true gap> idems := Idempotents(T);; gap> ForAll(idems, IsIdempotent); true # MatrixEntries: Test for Issue #164 gap> mat := [ > [Bipartition([[1, 2, 3, -2, -3], [-1]]), 0, 0, 0], > [0, Bipartition([[1, 3, -1], [2, -2, -3]]), 0, > Bipartition([[1, -1], [2, 3], [-2], [-3]])], > [0, 0, Bipartition([[1, 2, 3, -3], [-1], [-2]]), 0]]; [ [ <bipartition: [ 1, 2, 3, -2, -3 ], [ -1 ]>, 0, 0, 0 ], [ 0, <block bijection: [ 1, 3, -1 ], [ 2, -2, -3 ]>, 0, <bipartition: [ 1, -1 ], [ 2, 3 ], [ -2 ], [ -3 ]> ], [ 0, 0, <bipartition: [ 1, 2, 3, -3 ], [ -1 ], [ -2 ]>, 0 ] ] gap> R := ReesZeroMatrixSemigroup(PartitionMonoid(3), mat);; gap> MatrixEntries(R); [ 0, <bipartition: [ 1, 2, 3, -2, -3 ], [ -1 ]>, <bipartition: [ 1, 2, 3, -3 ], [ -1 ], [ -2 ]>, <block bijection: [ 1, 3, -1 ], [ 2, -2, -3 ]>, <bipartition: [ 1, -1 ], [ 2, 3 ], [ -2 ], [ -3 ]> ] gap> mat := [ > [Bipartition([[1, 2], [3, -1, -2], [-3]]), > Bipartition([[1, -2], [2, 3, -3], [-1]])], > [Bipartition([[1, 2, -1], [3], [-2, -3]]), > Bipartition([[1, 3, -1], [2, -2, -3]])], > [Bipartition([[1, 2, -2, -3], [3, -1]]), > Bipartition([[1, -1, -2], [2, 3, -3]])]]; [ [ <bipartition: [ 1, 2 ], [ 3, -1, -2 ], [ -3 ]>, <bipartition: [ 1, -2 ], [ 2, 3, -3 ], [ -1 ]> ], [ <bipartition: [ 1, 2, -1 ], [ 3 ], [ -2, -3 ]>, <block bijection: [ 1, 3, -1 ], [ 2, -2, -3 ]> ], [ <block bijection: [ 1, 2, -2, -3 ], [ 3, -1 ]>, <block bijection: [ 1, -1, -2 ], [ 2, 3, -3 ]> ] ] gap> R := ReesZeroMatrixSemigroup(PartitionMonoid(3), mat);; gap> MatrixEntries(R); [ <bipartition: [ 1, 2, -1 ], [ 3 ], [ -2, -3 ]>, <block bijection: [ 1, 2, -2, -3 ], [ 3, -1 ]>, <bipartition: [ 1, 2 ], [ 3, -1, -2 ], [ -3 ]>, <block bijection: [ 1, 3, -1 ], [ 2, -2, -3 ]>, <block bijection: [ 1, -1, -2 ], [ 2, 3, -3 ]>, <bipartition: [ 1, -2 ], [ 2, 3, -3 ], [ -1 ]> ] # IsomorphismReesMatrixSemigroup, infinite gap> IsomorphismReesMatrixSemigroup(FreeInverseSemigroup(2)); Error, the argument must be a finite simple semigroup # IsomorphismReesZeroMatrixSemigroup, infinite gap> IsomorphismReesZeroMatrixSemigroup(FreeSemigroup(2)); Error, the argument must be a finite 0-simple semigroup # IsomorphismReesZeroMatrixSemigroup, error, 1/1 gap> IsomorphismReesZeroMatrixSemigroup(RegularBooleanMatMonoid(2)); Error, the argument (a semigroup) is not a 0-simple semigroup # IsomorphismReesMatrixSemigroup: for a simple semigroup gap> S := SemigroupIdeal( > Semigroup([ > Bipartition([[1, 2, 3, 6, 7, 8, -2, -4, -5, -6], [4, 5, -1, -8], [-3], > [-7]]), > Bipartition([[1, 5, 8], [2, 7, -3, -6], [3, 4, -4, -7], [6, -1, -5], > [-2, -8]])]), > [Bipartition([[1, 2, 3, 4, 5, 6, 7, 8, -1, -2, -4, -5, -6, -8], [-3], > [-7]])]);; gap> IsomorphismReesMatrixSemigroup(S);; # IsomorphismReesMatrixSemigroup: for a 0-simple semigroup 1/2 gap> S := Semigroup([ > Transformation([1, 1, 5, 1, 3, 1, 9, 1, 7, 5]), > Transformation([1, 1, 2, 1, 4, 1, 6, 1, 8, 2]), > Transformation([1, 5, 1, 3, 1, 9, 1, 7, 1, 7])]);; gap> IsomorphismReesZeroMatrixSemigroup(S);; # IsomorphismReesMatrixSemigroup: for a 0-simple semigroup 2/2 gap> S := Semigroup([ > Transformation([1, 1, 5, 1, 3, 1, 9, 1, 7, 5]), > Transformation([1, 1, 2, 1, 4, 1, 6, 1, 8, 2]), > Transformation([1, 5, 1, 3, 1, 9, 1, 7, 1, 7])]);; gap> S := Semigroup(MultiplicativeZero(S), S);; gap> IsomorphismReesZeroMatrixSemigroup(S);; # IsomorphismReesMatrixSemigroup: for a non-simple or non-0-simple gap> S := Semigroup(Transformation([2, 1]), Transformation([2, 2]));; gap> IsomorphismReesMatrixSemigroup(S); Error, the argument (a semigroup) is not simple # IsomorphismReesZeroMatrixSemigroup, bug 1/1 gap> S := Semigroup(PartialPerm([1]), PartialPerm([])); <partial perm monoid of rank 1 with 2 generators> gap> IsomorphismReesMatrixSemigroup(S); Error, the argument (a semigroup) is not simple gap> IsomorphismReesZeroMatrixSemigroup(S);; gap> Size(Range(last)); 2 # ChooseHashFunction: Test for RZMS elements over pc group gap> G := ElementaryAbelianGroup(4);; gap> a := AsList(G)[1];; b := AsList(G)[2];; gap> mat := [[a, 0, b], [b, 0, 0], [0, a, b]];; gap> S := ReesZeroMatrixSemigroup(G, mat);; gap> x := RMSElement(S, 1, a, 2);; gap> func := ChooseHashFunction(x, 25531).func;; gap> data := ChooseHashFunction(x, 25531).data; [ 101, 25531 ] gap> x := RMSElement(S, 2, b, 3);; gap> func(x, data);; gap> x := MultiplicativeZero(S);; gap> func(x, data);; # ChooseHashFunction: Test for RZMS elements over transformation semigroups gap> G := FullTransformationMonoid(3);; gap> a := AsList(G)[1];; b := AsList(G)[2];; gap> mat := [[a, 0, b], [b, 0, 0], [0, a, b]];; gap> S := ReesZeroMatrixSemigroup(G, mat);; gap> x := RMSElement(S, 1, a, 2);; gap> func := ChooseHashFunction(x, 25531).func;; gap> data := ChooseHashFunction(x, 25531).data; 25531 gap> func(x, data);; gap> x := MultiplicativeZero(S);; gap> func(x, data);; gap> func := ChooseHashFunction(x, 25531).func;; gap> data := ChooseHashFunction(x, 25531).data;; gap> func(x, data);; # ChooseHashFunction: Test for RZMS elements over bipartition semigroups gap> G := BrauerMonoid(3);; gap> a := AsList(G)[1];; b := AsList(G)[2];; gap> mat := [[a, 0, b], [b, 0, 0], [0, a, b]];; gap> S := ReesZeroMatrixSemigroup(G, mat);; gap> x := MultiplicativeZero(S);; gap> func := ChooseHashFunction(x, 25531).func;; gap> data := ChooseHashFunction(x, 25531).data; 25531 gap> func(x, data);; gap> func := ChooseHashFunction(x, 25531).func;; gap> data := ChooseHashFunction(x, 25531).data;; gap> func(x, data);; # HashTables: Over a pc group gap> G := PcGroupCode(67665939, 32);; # SmallGroup(32, 2);; gap> AssignGeneratorVariables(G); gap> mat := > [[f1, 0, f3, f2, 0, f1, f5], > [f2, 0, 0, f5, f1, f2, f1], > [0, f1, f2, f1, f2, f4, f4], > [f1, f2, f3, f4, f5, 0, f1], > [f5, f1, 0, f2, f4, f5, f5], > [f1, f2, f3, f5, 0, 0, 0], > [f5, f1, 0, f2, f4, f5, f1]];; gap> S := ReesZeroMatrixSemigroup(G, mat);; gap> x := RMSElement(S, 1, f1, 1);; gap> ht := HTCreate(x);; gap> for x in S do > HTAdd(ht, x, true); > od; # ChooseHashFunction: Test for RZMS elements over a group we can't hash yet gap> F := FreeGroup("a", "b");; gap> G := F / [F.1 ^ 2, F.2 ^ 3, (F.1 * F.2) ^ 5];; gap> a := AsList(G)[1];; b := AsList(G)[2];; gap> mat := [[a, 0, b], [b, 0, 0], [0, a, b]];; gap> S := ReesZeroMatrixSemigroup(G, mat);; gap> x := MultiplicativeZero(S);; gap> func := ChooseHashFunction(x, 25531).func;; # RandomSemigroup gap> RandomSemigroup(IsReesMatrixSemigroup);; gap> RandomSemigroup(IsReesMatrixSemigroup, 2);; gap> RandomSemigroup(IsReesMatrixSemigroup, 2, 2);; gap> RandomSemigroup(IsReesMatrixSemigroup, 2, 2, Group(()));; gap> RandomSemigroup(IsReesMatrixSemigroup, "a"); Error, the 2nd argument (number of rows) must be a positive integer gap> RandomSemigroup(IsReesMatrixSemigroup, 2, "a"); Error, the 3rd argument (number of columns) must be a positive integer gap> RandomSemigroup(IsReesMatrixSemigroup, 2, 2, "a"); Error, the 4th argument must be a permutation group gap> RandomSemigroup(IsReesMatrixSemigroup, 2, 2, Group(()), 1); Error, expected at most 3 arguments, found 4 gap> RandomSemigroup(IsReesZeroMatrixSemigroup);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2, 2);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2, 2, Group(()));; gap> RandomSemigroup(IsReesZeroMatrixSemigroup, "a"); Error, the 2nd argument (number of rows) must be a positive integer gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2, "a"); Error, the 3rd argument (number of columns) must be a positive integer gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2, 2, "a"); Error, the 4th argument must be a permutation group gap> RandomSemigroup(IsReesZeroMatrixSemigroup, 2, 2, Group(()), 1); Error, expected at most 3 arguments, found 4 gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2, 2);; gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2, 2, > Group(()));; gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, "a"); Error, the 2nd argument (number of rows) must be a positive integer gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2, "a"); Error, the 3rd argument (number of columns) must be a positive integer gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2, 2, > "a"); Error, the 4th argument must be a permutation group gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 2, 2, > Group(()), 1); Error, expected at most 3 arguments, found 4 gap> RandomSemigroup(IsReesZeroMatrixSemigroup and IsRegularSemigroup, 3, 2);; # Test RMSElementNC gap> R := ReesMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), (1, 3), (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]); <Rees matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )> gap> x := RMSElementNC(R, 1, (1, 2), 1); (1,(1,2),1) gap> x in R; true # semirms: MultiplicativeZero, for a Rees 0-matrix semigroup, 1 gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]); <Rees 0-matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )> gap> R := Semigroup(R);; gap> MultiplicativeZero(R); 0 # semirms: MultiplicativeZero, for a Rees 0-matrix subsemigroup, 1 gap> R := ReesZeroMatrixSemigroup(Group(()), [[()]]);; gap> U := Semigroup(RMSElement(R, 1, (), 1));; gap> MultiplicativeZero(U); (1,(),1) gap> IsTrivial(U); true gap> R := ReesZeroMatrixSemigroup( > Semigroup([Transformation([1, 1]), Transformation([2, 2])]), > [[Transformation([1, 1])]]);; gap> U := Semigroup(RMSElement(R, 1, Transformation([1, 1]), 1)); <subsemigroup of 1x1 Rees 0-matrix semigroup with 1 generator> gap> MultiplicativeZero(U); (1,Transformation( [ 1, 1 ] ),1) # semirms: MultiplicativeZero, for a Rees 0-matrix subsemigroup, 2 gap> G := SymmetricGroup(3);; gap> R := ReesZeroMatrixSemigroup(G, > [[(), 0, (1, 2)], > [(), (1, 3, 2), (1, 2)], > [(1, 3), 0, (1, 3, 2)]]); <Rees 0-matrix semigroup 3x3 over Sym( [ 1 .. 3 ] )> gap> U := ReesZeroMatrixSubsemigroup(R, [1], G, [1, 3]); <subsemigroup of 3x3 Rees 0-matrix semigroup with 3 generators> gap> MultiplicativeZero(U); fail gap> U := ReesZeroMatrixSubsemigroup(R, [1, 3], G, [1]); <subsemigroup of 3x3 Rees 0-matrix semigroup with 3 generators> gap> MultiplicativeZero(U); fail gap> U := ReesZeroMatrixSubsemigroup(R, [1, 3], G, [1, 3]); <subsemigroup of 3x3 Rees 0-matrix semigroup with 4 generators> gap> MultiplicativeZero(U); fail gap> U := ReesZeroMatrixSubsemigroup(R, [3], G, [3]); <subsemigroup of 3x3 Rees 0-matrix semigroup with 2 generators> gap> MultiplicativeZero(U); fail gap> U := ReesZeroMatrixSubsemigroup(R, [1], Group(()), [1]); <subsemigroup of 3x3 Rees 0-matrix semigroup with 1 generator> gap> MultiplicativeZero(U); (1,(),1) # Test IsomorphismPermGroup gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]); <Rees 0-matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )> gap> IsomorphismPermGroup(R); Error, the argument (a semigroup) does not satisfy IsGroupAsSemigroup gap> S := Semigroup(MultiplicativeZero(R));; gap> IsomorphismPermGroup(S); <subsemigroup of 3x4 Rees 0-matrix semigroup with 1 generator> -> Group(()) gap> map := IsomorphismPermGroup(S); <subsemigroup of 3x4 Rees 0-matrix semigroup with 1 generator> -> Group(()) gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true gap> S := Semigroup(RMSElementNC(R, 1, (1, 3), 1));; gap> map := IsomorphismPermGroup(S);; gap> Range(map); Group([ (1,2) ]) gap> BruteForceIsoCheck(map); true gap> BruteForceInverseCheck(map); true # Test GroupOfUnits gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> GroupOfUnits(R); fail gap> S := Semigroup(MultiplicativeZero(R));; gap> GroupOfUnits(S); <subsemigroup of 3x4 Rees 0-matrix semigroup with 1 generator> gap> S := Semigroup(RMSElementNC(R, 1, (1, 3), 1));; gap> GroupOfUnits(S); <subsemigroup of 3x4 Rees 0-matrix semigroup with 2 generators> gap> S := Semigroup(RMSElementNC(R, 2, (1, 3), 3));; gap> GroupOfUnits(S); fail # Test Random gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> Random(R) in R; true # Test ViewString for a Rees 0-matrix semigroup ideal gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(1), [[()]]); <Rees 0-matrix semigroup 1x1 over Group(())> gap> R := Semigroup(GeneratorsOfSemigroup(R)); <subsemigroup of 1x1 Rees 0-matrix semigroup with 2 generators> gap> IsTrivial(MinimalIdeal(R)); true gap> MinimalIdeal(R); <trivial group> gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> I := SemigroupIdeal(R, MultiplicativeZero(R)); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsTrivial(I); true gap> I;; gap> I := SemigroupIdeal(R, MultiplicativeZero(R)); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsCommutative(I) and IsSimpleSemigroup(I); true gap> I;; gap> I := SemigroupIdeal(R, RMSElementNC(R, 2, (1, 3), 1)); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsCommutative(I); false gap> I; <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsZeroSimpleSemigroup(I); true gap> I; <0-simple regular Rees 0-matrix semigroup ideal with 1 generator> gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(), 0], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 4 ] )> gap> I := SemigroupIdeal(R, RMSElementNC(R, 2, (), 2)); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsInverseSemigroup(I); true gap> I; <inverse Rees 0-matrix semigroup ideal with 1 generator> gap> I := SemigroupIdeal(R, RMSElementNC(R, 2, (1, 3), 1), > RMSElementNC(R, 2, (1, 3), 3)); <regular Rees 0-matrix semigroup ideal with 2 generators> gap> I := SemigroupIdeal(R, RMSElementNC(R, 2, (1, 3), 3)); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[0, 0], [0, 0]]); <Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 4 ] )> gap> I := SemigroupIdeal(R, RMSElementNC(R, 2, (1, 3), 2)); <Rees 0-matrix semigroup ideal with 1 generator> gap> IsRegularSemigroup(I); false gap> I; <commutative non-regular Rees 0-matrix semigroup ideal with 1 generator> gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(), ()], [(), ()]]); <Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 4 ] )> gap> S := Semigroup(DClass(R, RMSElement(R, 1, (), 1))); <subsemigroup of 2x2 Rees 0-matrix semigroup with 96 generators> gap> IsSimpleSemigroup(S); true gap> I := SemigroupIdeal(S, S.1); <regular Rees 0-matrix semigroup ideal with 1 generator> gap> IsSimpleSemigroup(I); true gap> I; <simple Rees 0-matrix semigroup ideal with 1 generator> # Test MatrixEntries for an RMS gap> R := ReesMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), (1, 3), (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]); <Rees matrix semigroup 3x4 over Sym( [ 1 .. 4 ] )> gap> MatrixEntries(R); [ (), (3,4), (2,4,3), (1,2), (1,2,3,4), (1,2,4,3), (1,3,2), (1,3), (1,3)(2,4), (1,4,2) ] # Test GreensHClassOfElement for RZMS gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> H := GreensHClassOfElement(R, 1, 1); <Green's H-class: (1,(),1)> gap> IsGroupHClass(H); true gap> H := GreensHClassOfElement(R, 2, 3); <Green's H-class: (2,(),3)> gap> IsGroupHClass(H); false gap> H := GreensHClassOfElement(R, 3, 2); <Green's H-class: (3,(),2)> gap> IsGroupHClass(H); true # Test Idempotents gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> Idempotents(R); [ 0, (1,(1,2,3),1), (1,(),2), (1,(3,4),3), (1,(),4), (2,(),1), (2,(1,3)(2,4),2), (2,(2,3,4),4), (3,(1,2,4),1), (3,(1,4,3,2),2), (3,(1,3,4,2),3), (3,(1,2),4) ] gap> G := AsSemigroup(IsTransformationSemigroup, SymmetricGroup(4)); <transformation group of size 24, degree 4 with 2 generators> gap> R := ReesZeroMatrixSemigroup(G, [[IdentityTransformation, > IdentityTransformation]]); <Rees 0-matrix semigroup 2x1 over <transformation group of size 24, degree 4 with 2 generators>> gap> Idempotents(R); [ 0, (1,IdentityTransformation,1), (2,IdentityTransformation,1) ] gap> R := ReesZeroMatrixSemigroup(ZeroSemigroup(2), > [[Transformation([1, 1, 2]), Transformation([1, 1, 2])]]); <Rees 0-matrix semigroup 2x1 over <commutative non-regular transformation semigroup of size 2, degree 3 with 1 generator>> gap> Idempotents(R); [ (1,Transformation( [ 1, 1, 1 ] ),1), (2,Transformation( [ 1, 1, 1 ] ),1), 0 ] # IsIdempotentGenerated, for an RZMS gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), 0, (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> IsIdempotentGenerated(R); true gap> R := PrincipalFactor(DClass(FullTransformationMonoid(5), > Transformation([1, 1, 2, 3, 4])));; gap> IsIdempotentGenerated(R); true gap> R := ReesZeroMatrixSemigroup(SymmetricGroup(4), > [[(), 0], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Sym( [ 1 .. 4 ] )> gap> IsIdempotentGenerated(R); false # ReesZeroMatrixSubsemigroup is not a ReesZeroMatrixSemigroup gap> R := ReesZeroMatrixSemigroup(Group(()), [[(), 0], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Group(())> gap> S := Semigroup(Idempotents(R)); <subsemigroup of 2x2 Rees 0-matrix semigroup with 3 generators> gap> IsIdempotentGenerated(S); true # UnderlyingSemigroup is not IsGroup gap> R := ReesZeroMatrixSemigroup(Semigroup(Transformation([2, 1])), > [[Transformation([2, 1])]]);; gap> IsIdempotentGenerated(R); false gap> R := ReesZeroMatrixSemigroup(Semigroup(Transformation([2, 1])), > [[IdentityTransformation, IdentityTransformation], > [IdentityTransformation, Transformation([2, 1])]]);; gap> IsIdempotentGenerated(R); true # UnderlyingSemigroup is not IsGroupAsSemigroup gap> mat := [[ > Transformation([2, 3, 1]), > Transformation([2, 1]), > Transformation([1, 2, 1])]];; gap> R := ReesZeroMatrixSemigroup(FullTransformationMonoid(3), mat); <Rees 0-matrix semigroup 3x1 over <full transformation monoid of degree 3>> gap> IsIdempotentGenerated(R); false gap> R = Semigroup(Idempotents(R)); false gap> S := Monoid([ > Transformation([1, 2, 3, 1]), Transformation([2, 1, 3, 1]), > Transformation([2, 3, 3, 1]), Transformation([3, 2, 3, 1]), > Transformation([3, 3, 1, 2]), Transformation([3, 4, 1, 1]), > Transformation([4, 1, 2, 2]), Transformation([4, 2, 3, 3])]);; gap> R := ReesZeroMatrixSemigroup(S, [[IdentityTransformation]]);; gap> IsIdempotentGenerated(R); true # IsIdempotentGenerated, for an RMS gap> R := ReesMatrixSemigroup(SymmetricGroup(4), > [[(1, 3, 2), (), (1, 4, 2)], > [(), (1, 3)(2, 4), (1, 2, 3, 4)], > [(3, 4), (), (1, 2, 4, 3)], > [(), (2, 4, 3), (1, 2)]]);; gap> IsIdempotentGenerated(R); true # ReesMatrixSubsemigroup is not a ReesMatrixSemigroup gap> R := ReesMatrixSemigroup(SymmetricInverseMonoid(1), > [[PartialPerm([]), PartialPerm([])]]); <Rees matrix semigroup 2x1 over <symmetric inverse monoid of degree 1>> gap> S := Semigroup(RMSElement(R, 1, PartialPerm([1]), 1), > RMSElement(R, 2, PartialPerm([]), 1)); <subsemigroup of 2x1 Rees matrix semigroup with 2 generators> gap> IsIdempotentGenerated(S); false # UnderlyingSemigroup is not IsGroup gap> R := ReesMatrixSemigroup(Semigroup(Transformation([2, 1])), > [[Transformation([2, 1])]]); <Rees matrix semigroup 1x1 over <commutative transformation semigroup of degree 2 with 1 generator>> gap> IsIdempotentGenerated(R); false gap> R := ReesMatrixSemigroup(Semigroup(Transformation([2, 1])), > [[IdentityTransformation, IdentityTransformation], > [IdentityTransformation, Transformation([2, 1])]]); <Rees matrix semigroup 2x2 over <commutative transformation semigroup of degree 2 with 1 generator>> gap> IsIdempotentGenerated(R); true # UnderlyingSemigroup is not IsGroupAsSemigroup gap> mat := [[ > Transformation([2, 3, 1]), > Transformation([2, 1]), > Transformation([1, 2, 1])]];; gap> R := ReesMatrixSemigroup(FullTransformationMonoid(3), mat); <Rees matrix semigroup 3x1 over <full transformation monoid of degree 3>> gap> IsIdempotentGenerated(R); false gap> R = Semigroup(Idempotents(R)); false gap> S := Monoid([ > Transformation([1, 2, 3, 1]), Transformation([2, 1, 3, 1]), > Transformation([2, 3, 3, 1]), Transformation([3, 2, 3, 1]), > Transformation([3, 3, 1, 2]), Transformation([3, 4, 1, 1]), > Transformation([4, 1, 2, 2]), Transformation([4, 2, 3, 3])]);; gap> R := ReesMatrixSemigroup(S, [[IdentityTransformation]]); <Rees matrix semigroup 1x1 over <transformation monoid of degree 4 with 8 generators>> gap> IsIdempotentGenerated(R); true # Test Size for infinite RMS and RZMS gap> S := FreeSemigroup(2);; gap> R := ReesZeroMatrixSemigroup(S, [[S.1]]); <Rees 0-matrix semigroup 1x1 over <free semigroup on the generators [ s1, s2 ]>> gap> Size(R); infinity gap> R := ReesMatrixSemigroup(S, [[S.1]]); <Rees matrix semigroup 1x1 over <free semigroup on the generators [ s1, s2 ]>> gap> Size(R); infinity # Representative gap> R := ReesMatrixSemigroup(Group((1, 2)), [[(), (1, 2)], [(1, 2), ()]]);; gap> Representative(R); (1,(),1) # Pickling gap> filename := Filename(DirectoryTemporary(), "rms.p");; gap> R := ReesMatrixSemigroup(Group((1, 2)), [[(), (1, 2)], [(1, 2), ()]]); <Rees matrix semigroup 2x2 over Group([ (1,2) ])> gap> WriteGenerators(filename, [R]); IO_OK gap> RR := ReadGenerators(filename)[1]; <Rees matrix semigroup 2x2 over Group([ (1,2) ])> gap> Matrix(RR) = Matrix(R); true gap> UnderlyingSemigroup(RR) = UnderlyingSemigroup(R); true gap> R := ReesZeroMatrixSemigroup(Group((1, 2)), [[(), (1, 2)], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> WriteGenerators(filename, [R]); IO_OK gap> RR := ReadGenerators(filename)[1]; <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> Matrix(RR) = Matrix(R); true gap> UnderlyingSemigroup(RR) = UnderlyingSemigroup(R); true gap> R := ReesMatrixSemigroup(Group((1, 2)), [[(), (1, 2)], [(1, 2), ()]]); <Rees matrix semigroup 2x2 over Group([ (1,2) ])> gap> IO_Pickle(IO_File(filename, "r"), R); IO_Error gap> R := ReesZeroMatrixSemigroup(Group((1, 2)), [[(), (1, 2)], [(1, 2), ()]]); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> IO_Pickle(IO_File(filename, "r"), R); IO_Error gap> filename := "bananas"; "bananas" gap> IO_Unpicklers.RMSX(filename); IO_Error gap> IO_Unpicklers.RZMS(filename); IO_Error # Idempotents gap> mat := [[ > Transformation([2, 3, 1]), > Transformation([2, 1]), > Transformation([1, 2, 1])]];; gap> R := ReesZeroMatrixSemigroup(FullTransformationMonoid(3), mat); <Rees 0-matrix semigroup 3x1 over <full transformation monoid of degree 3>> gap> Idempotents(R); [ (1,Transformation( [ 1, 1, 1 ] ),1), (1,Transformation( [ 1, 1, 2 ] ),1), (1,Transformation( [ 2, 1, 2 ] ),1), (1,Transformation( [ 2, 2, 2 ] ),1), (1,Transformation( [ 3, 1, 1 ] ),1), (1,Transformation( [ 3, 1, 2 ] ),1), (1,Transformation( [ 3, 1, 3 ] ),1), (1,Transformation( [ 3, 2, 2 ] ),1), (1,Transformation( [ 3, 3, 2 ] ),1), (1,Transformation( [ 3, 3, 3 ] ),1), (2,Transformation( [ 1, 1, 1 ] ),1), (2,Transformation( [ 1, 1 ] ),1), (2,Transformation( [ 2, 1, 1 ] ),1), (2,Transformation( [ 2, 1, 2 ] ),1), (2,Transformation( [ 2, 1 ] ),1), (2,Transformation( [ 2, 2, 2 ] ),1), (2,Transformation( [ 2, 2 ] ),1), (2,Transformation( [ 2, 3, 3 ] ),1), (2,Transformation( [ 3, 1, 3 ] ),1), (2,Transformation( [ 3, 3, 3 ] ),1), (3,Transformation( [ 1, 1, 1 ] ),1), (3,Transformation( [ 1, 2, 1 ] ),1), (3,Transformation( [ 1, 2, 2 ] ),1), (3,Transformation( [ 2, 2, 2 ] ),1), (3,Transformation( [ 3, 2, 2 ] ),1), (3,Transformation( [ 3, 2, 3 ] ),1), (3,Transformation( [ 3, 3, 3 ] ),1), 0 ] gap> NrIdempotents(R) = Length(Idempotents(R)); true gap> NrIdempotents(R); 28 gap> S := ReesZeroMatrixSemigroup(Group([(1, 2)]), > [[(1, 2), (1, 2)], [0, ()]]); <Rees 0-matrix semigroup 2x2 over Group([ (1,2) ])> gap> Idempotents(S); [ 0, (1,(1,2),1), (2,(1,2),1), (2,(),2) ] gap> NrIdempotents(S) = Length(Idempotents(S)); true gap> NrIdempotents(S); 4 # gap> SEMIGROUPS.StopTest(); gap> STOP_TEST("Semigroups package: standard/semigroups/semirms.tst"); [Dauer der Verarbeitung: 0.69 Sekunden, vorverarbeitet 2026-05-01] |
2026-05-26