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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 ${\le }_S$ of $S$. Using the concept of an anticone, all partially ordered groups which are epimorphic images of a semigroup $(S,·,{\le }_S)$ 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 $(S,{\le }_S)$. Also a characterization by means of the structure...