Covers for monoids

Abstract

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 M and a morphism θ from M onto M such that M is an extension of a submonoid T by a group and θ maps T 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 Ash's 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.

A tribute to John B. Fountain

“…This involved a beautiful insight in constructing covers, which was used in other contexts by both John and others. In particular, after a month in Paris, John, Pin and Pascal Weil produced a monster paper on covers [23], which has not been surpassed in the area.…”

Soft E-inversive Semigroup

“…denote all weak inverses of s. [6] A Subsemigroup T of a E-inversive semigroup S is said to be unitary if for any , [6]. A subsemigroup T of E-inversive semigroup S is said to be noamal If T such that the following three conditions.…”

Musings around the geometry of interaction, and coherence

“…In other words, M arises as an extension of a group, G(M ), by an involutive inf-semi-lattice, E(M ) 2 , in the category of inverse monoids. While the reader may be more accustomed to the notion of group extensions, i.e., of extensions of groups by groups in the category of groups, the latter arises already in Fountain et al (2002); more information can be found in the latter reference.…”