Counter Example of Sandwich set

Question

• Let $S$ be a semigroup with set $E$ of idempotents, and let $e,f \in E$. Then the set $S(e,f) = \{ g \in V(ef) \cap E : \ \ ge = fg = g \}$, where $V(a)$ is the set consisting of all inverse of $a$, is called Sandwich set.

We know that $S(e,f)$ is nonempty if $S$ is regular semigroup. I want to find any counter example If $S$ is not regular and $\exists \ \ e,f \in E$ such that $S(e,f) = \varnothing$.

Any help would be appreciate . Thank you

Answer 1

Minimal counterexample. Take the semigroup $S = \{e, f, ef, 0\}$ with $e^2 = e$, $f^2 = f$, $fe = 0$ and $0$ is a zero of $S$. Then $ef$ is not regular and has no inverse. Thus $V(ef) = \emptyset$ and hence $S(e,f) = 0$.