Given a semigroup $S$, is there terminology for those $x \in S$ such that the following hold?
- $x$ is idempotent
- Given any idempotent $y \in S$, we have $xy=yx$.
Comments.
Let $E$ denote the set of idempotents of $S$. Then given any $x \in S$ with the above properties, the set $E$ is closed under the function $s \in S \mapsto xs \in S$.
The collection of all $x \in S$ with the above two properties forms a subsemilattice of $S$.
I do not know whether there is an established terminology, but if you really need a term, you may try $E$-central. An element is central if it commutes with every other element and the letter $E$ is usually used in reference to idempotents. For instance, the class of finite semigroups in which the idempotents commute is denoted by $\mathbf{Ecom}$ in the literature.