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