F-Inverse Monoids as Weakly Schreier Extensions
Peter F. Faul
TL;DR
The work investigates the structural characterization of $F$-inverse monoids via the weakly Schreier property of the natural extension $E(M)\to M\to M/\sigma$. It establishes that an inverse monoid is $F$-inverse iff this extension is weakly Schreier, enabling a description via relaxed factor systems. It shows that every $F$-inverse monoid is isomorphic to an almost semidirect product $F(Y,G)$ with $G = M/\sigma$ and $Y = E(M)$, connecting $F$-inverse monoids to almost actions. In the Clifford case, it gives a characterization through Artin gluings of frames: $\mathrm{Gl}(f)$ with $f:G\to Y$ satisfying $f(gh)\wedge f(g) = f(g)\wedge f(h)$ yields an $F$-inverse Clifford semigroup, and abelian quotients $M/\sigma$ exhaust this construction. Altogether the paper provides a unified framework for constructing and classifying $F$-inverse monoids via weakly Schreier extensions, relaxed factor systems, and gluing-style constructions.
Abstract
It is known that an inverse monoid $M$ is E-unitary if and only if the following diagram is an extension: $E(M) \to M \to M/σ$, where $E(M)$ is the semilattice of idempotents and $M/σ$ is the minimal group quotient. F-inverse monoids are another fundamental class of inverse semigroup and all F-inverse monoids are E-unitary. Thus given that F-inverse monoids have an associated extension it is natural to ask if these extensions satisfy any special properties. Indeed we show that $M$ is F-inverse if and only if the aforementioned extension is weakly Schreier. This latter result allows us to make use of relaxed factor systems to provide a new characterization of F-inverse monoids. We end by restricting to the Clifford case and find a new characterization of these with much in common with Artin gluings of frames.