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

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57 changes: 57 additions & 0 deletions gap/attributes/isomorph.gi
Original file line number Diff line number Diff line change
Expand Up @@ -327,3 +327,60 @@ 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, map, SSS, i, j;
# 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 := DigraphReflexiveTransitiveReduction(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, []);
for j in N[i] do
map := function(elm)
return idemp * elm;
end;
Add(H[i], MappingByFunction(L[j], L[i], map));
od;
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It seems like there is no need to keep re-creating the function map inside the for loop.

Suggested change
for j in N[i] do
map := function(elm)
return idemp * elm;
end;
Add(H[i], MappingByFunction(L[j], L[i], map));
od;
map := elm -> idemp * elm;
for j in N[i] do
Add(H[i], MappingByFunction(L[j], L[i], map));
od;
# Add(H[i], MappingByFunction(L[i], L[i], map)); # Add this if you do actually need `i` to be in `N[i]`

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
Original file line number Diff line number Diff line change
Expand Up @@ -102,3 +102,6 @@ DeclareProperty("IsSurjectiveSemigroup", IsSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsRegularSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsMonoidAsSemigroup);
InstallTrueMethod(IsSurjectiveSemigroup, IsIdempotentGenerated);

DeclareProperty("IsOrthogroup", IsSemigroup);
DeclareSynonym("IsOrthoGroup", IsOrthogroup);
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Suggested change
DeclareSynonym("IsOrthoGroup", IsOrthogroup);
DeclareSynonymAttr("IsOrthoGroup", IsOrthogroup);

6 changes: 6 additions & 0 deletions gap/attributes/properties.gi
Original file line number Diff line number Diff line change
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);