Term for $eSe$ where $S$ is a semigroup and $e \in S$ is an idempotent

Question

For a (possibly non-unital) ring $R$ and an idempotent $e \in R$, $eRe$ is a unital ring with identity $e$ and is known as a corner ring.

Now, given any semigroup $S$ and any idempotent $e \in S$, $eSe$ is a monoid with identity $e$.

But what is the name for a monoid of the form $eSe$? Is it called a "corner monoid", analogous to "corner ring"?

Answer 1

In Mathematical Foundations of Automata Theory by Jean-´Eric Pin, pp.36

The semigroup $eSe$ $[eS, Se]$ is called the [left, right ] local semigroup associated with $e$.