The idempotent problem for an inverse monoid

We generalize the word problem for groups, the formal
language of all words in the generators that represent the identity,
to inverse monoids. In particular, we introduce the \emph{idempotent
problem}, the formal language of all words representing
idempotents, and investigate how the properties of an inverse monoid
are related to the formal language properties of its idempotent problem.
We show that if an inverse monoid is either E-unitary
or has a finite set of idempotents, then its idempotent problem is
regular if and only if the inverse monoid is finite. We
also give examples of inverse monoids with context-free idempotent
problems, including all Bruck-Reilly extensions of finite groups.