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Decomposition of some Completely Regular Semigroups into Strong Semilattices of Semigroups #731

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Decomposition of some Completely Regular Semigroups into Strong Semilattices of Semigroups #731

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cff1a6a
Add IsomorphismSemigroup from Clifford smgp to SSS
tomcontileslie May 19, 2021
79d230f
Add IsOrthoGroup property
tomcontileslie May 19, 2021
f8a4366
SSS Decomp bugfix
tomcontileslie May 26, 2021
2c1d259
SSS Decomp: add test
tomcontileslie May 26, 2021
21e5eae
SSS Decomp: add 2 more tests
tomcontileslie May 26, 2021
825aee3
SSS Decomp: address Wilf's comments
tomcontileslie May 26, 2021
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58 changes: 58 additions & 0 deletions gap/attributes/isomorph.gi
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Expand Up @@ -327,3 +327,61 @@ function(S)
UseIsomorphismRelation(H, G);
return H;
end);

InstallMethod(IsomorphismSemigroup,
"for IsStrongSemilatticeOfSemigroups and a Clifford semigroup",
[IsStrongSemilatticeOfSemigroups, IsSemigroup and IsFinite],
function(filt, S)
local A, idemps, n, D, N, L, classes, idemp, DC, H, SSS, i, j, addfunc;
# decomposes a finite Clifford semigroup S into a strong semilattice of
# groups and returns an SSS object.
if not (IsCliffordSemigroup(S) and IsFinite(S)) then
TryNextMethod();
fi;
# There should be one idempotent per D-class, i.e. per semilattice element
# since the semilattice decomposition is by J-classes, and J = D here
A := Semigroup(Idempotents(S));
idemps := Elements(A);
n := Size(idemps);

# create semilattice
D := Digraph(NaturalPartialOrder(A));
# currently wrong way round
D := DigraphReverse(D);
N := OutNeighbours(D);
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You can directly get this via

Suggested change
N := OutNeighbours(D);
N := ReverseNaturalPartialOrder(A);

(although I don't think that necessarily (or at all?) contains x in N[x] for each x - if those are actually necessary, you will still want to add them or modify your code below to acts as if they are there).

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Thanks @wilfwilson, unfortunately I get a "no method found" error when running ReverseNaturalPartialOrder on some inputs - for example:

gap> S := Semigroup(Transformation([1, 2, 4, 5, 6, 3, 7, 8]), Transformation([3, 3, 4, 5, 6, 2, 7, 8]), Transformation([1, 2, 5, 3, 6, 8, 4, 4]));
<transformation semigroup of degree 8 with 3 generators>
gap> IsCliffordSemigroup(S);
true
gap> A := Semigroup(Idempotents(S));
<transformation monoid of degree 8 with 3 generators>
gap> NaturalPartialOrder(A);
[ [ 2, 3, 4 ], [ 4 ], [ 4 ], [  ] ]
gap> ReverseNaturalPartialOrder(A);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 2nd choice method found for `ReverseNaturalPartialOrder' on 1 arguments at /Applications/GAP/lib/methsel2.g:250 called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
 called from read-eval loop at *stdin*:18
type 'quit;' to quit to outer loop

Although you are correct that we don't need x in N[x] since the homomorphisms in this case are the identity, and the strong semilattice constructor is clever enough to fill that in upon creation.

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Also, I'm realising that taking the reflexive transitive reduction of D is redundant, since the SSS constructor effectively reverts this, so I'll remove that part


# populate list of semigroups in semilattice.
# keep a list of D-classes at the same time, to figure out where elements are
L := [];
classes := [];
for i in [1 .. n] do
idemp := idemps[i]; # the idempotent of this D-class
DC := DClass(S, idemp);
Add(L, Semigroup(DC));
Add(classes, DC);
od;

# populate list of homomorphisms
H := [];
for i in [1 .. n] do
idemp := idemps[i];
Add(H, []);
addfunc := function(x, a, b, c) # horrible namespace hack (credits JDM)
Add(x, [a, b, z -> c * z]);
end;
for j in N[i] do
addfunc(H[i], L[j], L[i], idemps[i]);
od;
Apply(H[i], x -> MappingByFunction(x[1], x[2], x[3]));
od;

SSS := StrongSemilatticeOfSemigroups(D, L, H);

return MagmaIsomorphismByFunctionsNC(S,
SSS,
x -> SSSE(SSS,
Position(classes,
DClass(S, x)),
x),
x -> x![3]);
end);
3 changes: 3 additions & 0 deletions gap/attributes/properties.gd
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Expand Up @@ -102,3 +102,6 @@ DeclareProperty("IsSurjectiveSemigroup", IsSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsRegularSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsMonoidAsSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsIdempotentGenerated);

DeclareProperty("IsOrthogroup", IsSemigroup);
DeclareSynonymAttr("IsOrthoGroup", IsOrthogroup);
6 changes: 6 additions & 0 deletions gap/attributes/properties.gi
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Expand Up @@ -1784,3 +1784,9 @@ x -> UnderlyingSemigroupOfSemigroupWithAdjoinedZero(x) <> fail);
InstallMethod(IsSurjectiveSemigroup, "for a semigroup",
[IsSemigroup],
S -> IsEmpty(IndecomposableElements(S)));

InstallMethod(IsOrthogroup, "for a semigroup",
[IsSemigroup],
function(S)
return IsCompletelyRegularSemigroup(S) and IsOrthodoxSemigroup(S);
end);
63 changes: 63 additions & 0 deletions tst/standard/isomorph.tst
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Expand Up @@ -385,9 +385,72 @@ gap> G := AutomorphismGroup(S);
gap> StructureDescription(G);
"S3"

# IsomorphismSemigroup for Clifford semigroups to strong semilattices (1/3)
gap> S := DualSymmetricInverseMonoid(5);;
gap> T := IdempotentGeneratedSubsemigroup(S);;
gap> map := IsomorphismSemigroup(IsStrongSemilatticeOfSemigroups, T);;
gap> S := Range(map);;
gap> S;
<strong semilattice of 52 semigroups>
gap> IsStrongSemilatticeOfSemigroups(S);
true
gap> failed := [];;
gap> for x in T do
> if x <> (x ^ map)![3] then
> Add(failed, x);;
> fi;
> od;
gap> failed;
[ ]
gap> IsomorphismSemigroups(S, T) <> fail;
true
gap> SemilatticeOfStrongSemilatticeOfSemigroups(S);
<immutable meet semilattice digraph with 52 vertices, 358 edges>
gap> BruteForceIsoCheck(map);
true

# IsomorphismSemigroup for Clifford semigroups to strong semilattices (2/3)
gap> T := Semigroup(Transformation([1, 2, 4, 5, 6, 3, 7, 8]),
> Transformation([3, 3, 4, 5, 6, 2, 7, 8]),
> Transformation([1, 2, 5, 3, 6, 8, 4, 4]));;
gap> Size(T);
864
gap> NrIdempotents(T);
4
gap> S := IsomorphismSemigroup(IsStrongSemilatticeOfSemigroups, T);;
gap> map := IsomorphismSemigroup(IsStrongSemilatticeOfSemigroups, T);;
gap> S := Range(map);
<strong semilattice of 4 semigroups>
gap> Size(S);
864
gap> # Property of CLifford semigroups:
gap> List(SemigroupsOfStrongSemilatticeOfSemigroups(S), IsGroupAsSemigroup);
[ true, true, true, true ]
gap> BruteForceIsoCheck(map);
true

# IsomorphismSemigroup for Clifford semigroups to strong semilattices (3/3)
gap> T := Semigroup(List([2 .. 10],
> x -> Transformation(ListWithIdenticalEntries(x, 1))));
<transformation semigroup of degree 10 with 9 generators>
gap> IsCliffordSemigroup(T);
true
gap> map := IsomorphismSemigroup(IsStrongSemilatticeOfSemigroups, T);;
gap> S := Range(map);
<strong semilattice of 9 semigroups>
gap> BruteForceIsoCheck(map);
true
gap> Size(S) = Size(T);
true
gap> SemilatticeOfStrongSemilatticeOfSemigroups(S) =
> DigraphReflexiveTransitiveClosure(ChainDigraph(9));
true

# SEMIGROUPS_UnbindVariables
gap> Unbind(map);
gap> Unbind(S);
gap> Unbind(T);
gap> Unbind(x);

#
gap> SEMIGROUPS.StopTest();
Expand Down