Is there a name for a semigroup whose idempotents form a subsemigroup?

Question

For a semigroup $S,$ I will denote by $E(S)$ the set of all idempotents of $S$. For $X\subseteq S,$ let $X^2$ mean $\{xy\,|\,x,y\in X\}.$

Is there a name for the class of semigroups $S$ such that $$\left(E(S)\right)^2\subseteq E(S)?$$

To have an example, in every inverse semigroup, the idempotents form a subsemigroup. More generally, as rschwieb points out in a comment, any semigroup such that the idempotents commute with each other satisfies this condition.

I need a name to be able to search for information about such semigroups. So any contribution besides the name will be welcomed.

Answer 1

According to wikipedia, we have

• A regular semigroup whose idempotents forms a subsemigroup is called an orthodox semigroup.
• A completely regular semigroup whose idempotents forms a subsemigroup is called an orthogroup.

Every semigroup can be embedded into a regular semigroup. Perhaps subsemigroup of an orthodox semigroup comes closest to the condition you are looking for. I don't know whether there exists a semigroup which satisfies your condition without being a subsemigroup of an orthodox semigroup.