A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S * of S and a unary operation * in S satisfying: (1) x * ∈ S * ⋂ V_ S * (x) for all x∈ S; (2) (x * ) * =x for all x∈ S * ; (3) (x * y) * =y * x ** and (xy * ) * =y ** x * for all x,y∈ S, then S * is called a regular *- transversal of S ; if (3) is replaced with (xy) * =y * x * for all x,y∈ S, then S * is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.