F-inverse monoids as algebraic structures in enriched signature

Every $F$-inverse monoid can be equipped with the unary operation which maps each element to maximum of its $\sigma$-class. In this enriched signature, class all monoids forms a variety algebraic structures. We describe universal objects in several classes monoids, particular free monoids. More precisely, for every $X$-generated group $G$ we initial object category $F$ $F/\sigma=G$.