Fundamental Groups, Inverse Sch Utzenberger Automata, and Monoid Presentations

This paper gives decidable conditions for when a nitely generated subgroup of a free group is the fundamental group of a Sch utzenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to speciic types of inverse monoids as well as to monoids which are \nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups. This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a nite strongly connected inverse automaton is a Sch utzenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to \nearly inverse" monoids. We show that it is undecidable whether a nite strongly connected inverse automaton is a Sch utzenberger automaton of a monoid presentation of an E-unitary inverse monoid.