Example of a locally inverse semigroup which isn't a generalized inverse semigroup

Question

I'm studying Howie's Fundamentals of Semigroup Theory.

A semigroup $S$ is locally inverse if $eSe$ is inverse for any idempotent $e$ of $S$. A semigroup is a generalized inverse semigroup if is regular and its idempotents are a normal band.

It's very easy to show that any generalized inverse semigroup is locally inverse. Howie also comments that the reciprocal is not true. What would be a concrete example?

Answer 1

Take the semigroup $S = \{a, b, c, ab, 0\}$ where $a$, $b$ and $c$ are idempotent and $ca = c$, $ac = a$, $bc = c$, $cb = b$, $abc = a$, $ba = 0$. The non-zero elements form a $\mathcal{D}$-class:

\begin{align} \hline |{}^*a &\mid ab| \\ \hline |{}^*c &\mid{}^*b|\\ \hline \end{align}

This semigroup is regular, locally inverse, but the product of the idempotents $a$ and $b$ is not idempotent.