Strong divisibility commutative elements

We previously mentioned that semigroups can be strongly divisibility commutative. This occurs when the left and right principal ideals of the semigroup coincide for each element. In the case that the entire semigroup is not strongly divisibility commutative, there is a subset of the elements that are. So we will need to briefly examine strongly divisibility commutative elements and what they mean for Green's relations.

Definition. an element of a semigroup is strongly divisibility commutative if its left and right principal ideals coincide. The strong divisibility center consits of all elements that are strogly divisibility commutative.

It is trivial to see that central elements are strongly divisibility commutative, and therefore the center is a subset of the strongly divisibility center. Now consider the effect of strong divisibility upon the Green's relations. If two elements are strongly divisibility commutative, then they divisibility commute in the sense that L and R for the two elements are logically equivalent.

Theorem. if a and b are both strongly divisibility commutative then a L b is logically equivalent to a R b.

Proof. this will be proved by demonstrating implication in both directions. Suppose that a L b then we know that L(a) = L(b) but by strong divisibility commutative of elements we have R(a) = L(a) and L(b) = R(b) which implies R(a) = L(a) = L(b) = R(b) which by transitivity means that R(a) = R(b) which by definition means a R b. In the opposite direction a R b implies that R(a) = R(b) which implies that L(a) = R(a) = R(b) = L(b) which similarily implies L(a) = L(b) which by definition means a L b. So the proof of implication in both directions implies logical equivalence.

I managed to use this and Green's theorem to prove that idempotent-central semigropus (which include Clifford semigroups) are L,R idempotent separating. The H classes are also idempotent separating by Green's theorem, but there is no obvious way to get that the D classes are idempotent separating so we can't prove that.

Theorem. idempotent-central semigroups are L,R idempotent separating

Proof. let a and b be two idempotents of the semigroup, now a and b are both in the center by definition so they are strongly divisibility commutative. This implies that if a L b then a R b and that if a R b then a L b, so if idempotents are L related or R related then they are both L and R related, which would mean that they are H related. But by Green's theorem two idempotents cannot be H related, which would be a contradiction. So L and R must be idempotent separating.