Covers for monoids

A monoid M is an extension of a submonoid T by a group G if there is a morphism from M onto G such that T is the inverse image of the identity of G. Our first main theorem gives descriptions of such extensions in terms of groups acting on categories. The theory developed is also used to obtain a second main theorem which answers the following question. Given a monoid M and a submonoid T, under what conditions can we find a monoid and a morphism θ from onto M such that is an extension of a submonoid by a group and θ maps isomorphically onto T. These results can be viewed as generalisations of two seminal theorems of McAlister in inverse semigroup theory. They are also closely related to Ashʹs celebrated solution of the Rhodes conjecture in finite semigroup theory. McAlister proved that each inverse monoid admits an E-unitary inverse cover and gave a structure theorem for E-unitary inverse monoids. Many researchers have extended one or both of these results to wider classes of semigroups. Almost all these generalisations can be recovered from our two main theorems.