F-Inverse Monoids as Weakly Schreier Extensions

TL;DR

This paper characterizes F-inverse monoids as weakly Schreier extensions within inverse monoid theory, providing a new perspective and tools for understanding their structure, especially in the Clifford case.

Contribution

It establishes that F-inverse monoids correspond exactly to weakly Schreier extensions, offering a novel characterization using relaxed factor systems.

Findings

F-inverse monoids are characterized by weakly Schreier extensions.

A new characterization of Clifford F-inverse monoids is provided.

Connections to Artin gluings of frames are explored.

Abstract

It is known that an inverse monoid $M$ is E-unitary if and only if the following diagram is an extension: $E(M) \to M \to M/\sigma$, where $E(M)$ is the semilattice of idempotents and $M/\sigma$ 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.