F-Inverse Monoids as Weakly Schreier Extensions

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.