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∪ Ḡ) of G, with respect to the extended generating set X∪ Ḡ, where Ḡ is a bijective copy of G which encodes the m-operation in $F(G)$. The construction relies on a certain closure operator on the semilattice of all finite and connected subgraphs containing the origin of the Cayley graph Cay(G, X∪ Ḡ) and leads to a new and simpler proof of the universal property of $F(G)$.