A semigroup S is called a generalized F-semigroup if there exists a group congruence on S such that the identity class contains a greatest element with respect to the natural partial order ≤S of S. Using the concept of an anticone, all partially ordered groups which are epimorphic images of a semigroup (S, ·, ≤S) are determined. It is shown that a semigroup S is a generalized F-semigroup if and only if S contains an anticone, which is a principal order ideal of (S, ≤S). Also a characterization by means of the structure of the set of idempotents or by the existence of a particular element in S is given. The generalized Fsemigroups in the following classes are described: monoids, semigroups with zero, trivially ordered semigroups, regular semigroups, bands, inverse semigroups, Clifford semigroups, inflations of semigroups, and strong semilattices of monoids.