Decomposition of some Completely Regular Semigroups into Strong Semilattices of Semigroups

[tomcontileslie]

May 2021 update: this PR has only ticked one of four boxes, but it works on Clifford semigroups. Implementing IsomorphismSemigroup for other types of completely regular semigroups can be done in a different PR.

This PR introduces methods for decomposing certain completely regular semigroups into strong semilattices, as suggested in Issue #671. The aim is introduce IsomorphismSemigroup methods from the following types of semigroups:

• Clifford semigroups
• Normal orthogroups
• Normal cryptogroups
• Normal bands (?)

to SSS objects.

[wilfwilson]

Suggested change

	DeclareSynonym("IsOrthoGroup", IsOrthogroup);
	DeclareSynonymAttr("IsOrthoGroup", IsOrthogroup);

[wilfwilson]

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).

[tomcontileslie]

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.

[tomcontileslie]

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

[wilfwilson]

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]`