Application of verification techniques to inverse monoids

Abstract

The word problem for inverse monoids generated by
a set $Gamma$ subject to relations of the form $e=f$, where $e$ and $f$
are both idempotents in the free inverse monoid generated by $Gamma$,
is investigated. It is
shown that for every fixed monoid of this form the word problem
can be solved in polynomial time which solves an open problem of
Margolis and Meakin. For the uniform word problem, where the presentation is
part of the input, EXPTIME-completeness is shown.
For the Cayley-graphs of these
monoids, it is shown that the first-order theory with regular path
predicates is decidable. Regular path predicates allow to state
that there is a path from a node $x$ to a node $y$ that is labeled
with a word from some regular language. As a corollary, the decidability
of the generalized word problem is deduced. Finally, some results
on free partially commutative inverse monoids are presented.