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

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.
