\(F\)–semigroups

Abstract

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.