On Special Inverse Monoids with the Strong $F$-Inverse Property

An inverse monoid $S$ is called $F$-inverse if each $\sigma$-class of $S$, where $\sigma$ is the minimum group congruence of $S$, has a maximum element with respect to the natural order of $S$. Since the property of an inverse monoid being $F$-inverse immediately implies that it must be $E$-unitary, it follows that every $X$-generated $F$-inverse monoid with canonical maximum group image $G$ must be isomorphic to a quotient of the Margolis-Meakin expansion $M(G,X)$. If this is realised in such a way that all the maximal elements of each $\sigma$-class of $M(G,X)$ get identified, thus producing the top element of the corresponding $\sigma$-class of $S$, we say that $S$ is strongly $F$-inverse. Consequently, there is a universal $X$-generated inverse monoid $M_{sF}(G,X)$ with maximum group image $G$ and the strongly $F$-inverse property. We provide a presentation for this inverse monoid and show it can be further simplified upon introducing additional assumptions on the group $G$ (which will include all one-relator groups). We use this to provide a full description of all one-relator special inverse monoids with a cyclically reduced relator word that are strongly $F$-inverse. We also discuss some further examples and non-examples.