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 (·, 1, ⁻¹, ^ m) ( · , 1 , - 1 , 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 -Schützenberger graphs and P -expansions.