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.