Strongly divisibility commutative semigroups

We previously addressed the idea of divisibility commutative semigroups. These are semigroups in which the Green's L relation and the Green's R relation coincide. But it is apparent, even from the simplest case of the non-commutative divisibility commutative semigroup on three elements, that having the Green's L relation and the Green's R relation coincide does not imply that left and right principal ideals are going to be the same.

Definition. a semigroup is called strongly divisibility commutative if left and right ideals coincide.

The first thing is to prove that these strongly divisibility commutative semigroups are in fact divisibility commutative, for the sake of formality.

Theorem. strongly divisibility commutative semigroups are divisibility commutative

Proof. suppose that the L class of x and the L class of y are equal, then Sx = Sy, so since Sx = xS and Sy = yS we have that xS = Sx = Sy = yS which implies that xS = yS by the transitivity of multiplication. This means that x,y are in the same R class. In the reverse direction, if x,y are in the same R class then Sx = xS = yS = Sy implies that x,y are in the same L class. So R = L and the two Green's relations coincide when the principal ideals coincide.

It is trivial that commutative semigroups are strongly divisibility commutative. It is also the case that groups are strongly divisibility commutative. The interesting thing, is that there is a specific method to get the element that produces a given output element in the other principal ideal: namely conjugation of group elements.

Theorem. in a group if gx = yg is an element in both the left and right principal ideals of a given element g then x,y are conjugates of one another

Proof. if we take gx = yg and we multiply each side by g^(-1) to the front then we get x = g^(-1)yg which means that x is a conjugate of y. In the other direction, if we multiply each side by g^(-1) to the back we get gxg^(-1) = y and that means that x and y conjugates.

So conjugation in group theory actually is the basis of the strong divisibility commutativity. It is not hard to see then, why it is the case that self-conjugate elements are centered in groups, because conjugates are the elements that produce a given result in the opposite direction, and when this result is always the element must commute. It is known that in group theory, conjugates are directly related to commutativity in that for example the size of a conjugacy class is the order of the group divided by the commuting degree.

Previously we talked about Clifford semigroups, which are known to be the only divisibility commutative regular semigroups. According to the encyclopedia of mathematics, Clifford semigroups are strongly divisibility commutative as well because L and R classes coincide. I am willing to conjecture that since Clifford semigroups are completely regular, the element that produces a given output with respect to an element in the opposite argument order is the internal conjugate of that element in its H class. But I don't have a proof or a counter-example yet. The simpler case of groups is illuminating nonetheless.