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* is an element of S* boolean AND V-S* (x) for all x is an element of S (2) (x*)* = x for all X is an element of S*; (3) (x*y)* = y*x** and (xy*)* = y**x* for all x,y is an element of S, then S* is called a regular *-transversal of S: if (3) is replaced with (xy)* = y*x* for all x,y is an element of 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.