$F$–semigroups

A semigroup $S$ is called $F$- semigroup if there exists a group-congruence $\rho$ on $S$ such that every $\rho$-class contains a greatest element with respect to the natural partial order $\leq_S$ 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 $(S,\leq_S)$ 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.