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 Formula: see text and a class of partial actions of Formula: see text, determined by a set, Formula: see text, of identities, we define Formula: see text to be the universal Formula: see text-generated two-sided restriction monoid with respect to partial actions of Formula: see text determined by Formula: see text. This is an Formula: see text-restriction monoid which (for a certain Formula: see text) generalizes the Birget–Rhodes prefix expansion Formula: see text of a group Formula: see text. Our main result provides a coordinatization of Formula: see text via a partial action product of the idempotent semilattice Formula: see text of a similarly defined inverse monoid, partially acted upon by Formula: see text. The result by Fountain, Gomes 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 Formula: see text agree well with suitable properties of Formula: see text, such as being cancellative or embeddable into a group. We observe that if Formula: see text is an inverse monoid, then Formula: see text, the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion Formula: see text. This gives a presentation of Formula: see text and leads to a model for Formula: see text in terms of the known model for Formula: see text.