An inverse monoid $M$ is $F$-inverse if every $\sigma$-class $a\sigma$ admits a greatest element $a^\mathfrak{m}$ with respect to the natural partial order on $M$ ($\sigma$ denotes the smallest group congruence on $M$). An $F$-inverse monoid is necessarily $E$-unitary and such structures have attracted considerable attention. I intend to discuss the following topics:

$F$-inverse monoids with enriched signature: as algebraic structures of type $(2,1,1,0)$ (the second unary operation being $a\mapsto a^\mathfrak{m}$) the class of $F$-inverse monoids forms a variety (M. Kinyon). I shall present models of free $F$-inverse monoids and, more generally, universal objects in similarly defined categories (joint work with G. Kudryavtseva and M. B. Szendrei).

status of the Henckell--Rhodes problem (Does every finite inverse monoid admit a finite $F$-inverse cover?)

a possible extension of the Henckell--Rhodes problem (Is there an expansion from the category of finite inverse monoids to the category of finite $F$-inverse monoids?)