Inverse monoids and rational Schreier subsets of the free group

An inverse monoid M is an idempotent-pure image of the free inverse monoid on a set X if and only if M has a presentation of the form M=Inv<X:e_o=f_i, i∈I>, where e _i, f _i are idempotents of the free inverse monoid: every inverse monoid is an idempotent-separating image of one of this type. If R is an R-class of such an inverse monoid, then R may be regarded as a Schreier subset of the free group on X. This paper is concerned with an examination of which Schreier subsets arise in this way. In particular, if I is finite, then R is a rational Schreier subset of the free group. Not every rational Schreier set arises in this way, but every positively labeled rational Schreier set does.