A semigroup S is called F- semigroup if there exists a group-congruence ?? on S such that every ??-class contains a greatest element with respect to the natural partial order ???S of S (see [8]). This generalizes the concept of F-inverse semigroups introduced by V. Wagner [12] and investigated in [7]. Five different characterizations of general F-semigroups S are given: by means of residuals, by special principal anticones, by properties of the set of idempotents, by the maximal elements in (S,???S) and finally, an axiomatic one using an additional unary operation. Also F-semigroups in special classes are considered; in particular, inflations of semigroups and strong semilattices of monoids are studied.