Two-sided expansions of monoids

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget-Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid M and a class of partial actions of M, determined by a set, R, of identities, we define FRR(M) to be the universal M-generated two-sided restriction monoid with respect to partial actions of M determined by R. This is an F-restriction monoid which (for a certain R) generalizes the Birget-Rhodes prefix expansion (G) over tilde (R) of a group G. Our main result provides a coordinatization of FRR(M) via a partial action product of the idempotent semilattice E(FIR(M)) of a similarly defined inverse monoid, partially acted upon by M. The result by Fountain, Comes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of FRR(M) agree well with suitable properties of M, such as being cancellative or embeddable into a group. We observe that if M is an inverse monoid, then FIRs (M), the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson-Margolis-Steinberg generalized prefix expansion M-pr. This gives a presentation of M-pr and leads to a model for FRRs (M) in terms of the known model for M-pr.