A new approach to universal $F$-inverse monoids in enriched signature

We show that the universal $X$-generated $F$-inverse monoid $F(G)$, where $G$ is an $X$-generated group, introduced by Auinger, Szendrei and the first-named author, arises as a quotient inverse monoid of the Margolis-Meakin expansion $M(G,X\cup \bar{G})$ of $G$, with respect to the extended generating set $X\cup \bar{G}$, where $\bar{G}$ is a bijective copy of $G$ which encodes the $m$-operation in $F(G)$. The construction relies on a certain dual-closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph $Cay(G,X\cup \bar{G})$ and leads to a new and simpler proof of the universal property of $F(G)$.