$E$-unitary and $F$-inverse monoids, and closure operators on group Cayley graphs

We show that the category of $X$-generated $E$-unitary inverse monoids with greatest group image $G$ is equivalent to the category of $G$-invariant, finitary closure operators on the set of connected subgraphs of the Cayley graph of $G$. Analogously, we study $F$-inverse monoids in the extended signature $(\cdot, 1, ^{-1}, ^\mathfrak{m})$, and show that the category of $X$-generated $F$-inverse monoids with greatest group image $G$ is equivalent to the category of $G$-invariant, finitary closure operators on the set of all subgraphs of the Cayley graph of $G$. As an application, we show that presentations of $F$-inverse monoids in the extended signature can be studied by tools analogous to Stephen's procedure in inverse monoids, in particular, we introduce the notions of $F$-Sch\"utzenberger graphs and $P$-expansions.