F-regular semigroups

A semigroup S is called F-regular if S is regular and if there exists a group congruence ρ on S such that every ρ-class contains a greatest element with respect to the natural partial order of S (see [K.S. Nambooripad, Proc. Edinburgh Math. Soc. 23 (1980) 249–260]). These semigroups were investigated in [C.C. Edwards, Semigroup Forum 19 (1980) 331–345] where a description similar to the F-inverse case (see [R. McFadden, L. OʹCarroll, Proc. London Math. Soc. 22 (1971) 652–666]) is given. Further characterizations of F-regular semigroups, including an axiomatic one, are provided. The main objective is to give a new representation of such semigroups by means of Szendrei triples (see [M. Szendrei, Acta Sci. Math. 51 (1987) 229–249]). The particular case of F-regular semigroups S satisfying the identity (xy)*=y*x*, where x* S denotes the greatest element of the ρ-class containing x S, is considered. Also the F-inversive semigroups, for which this identity holds, are characterized.