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.
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