Structure of regular semigroups with a quasi-ideal inverse transversal

Abstract

Let S be a regular semigroup, and E(S) denotes the set of idempotents of S. An inverse subsemigroup S ~ of S is called an inverse transversal of S, if IS ~ n V(x) l = I for each x ~ S, where V(x) denotes the set of inverses of x. An inverse of x c S which belongs to S ~ is denoted by x ~ and x ~176 denotes (x~ ~ = (x~ -z. A subset Q of S is called a quasi-ideal of S if QSQ ~ Q. Let S be a regular semigroup with an inverse transversal S ~ that is a quasi-ideal of S. In [4], it has been shown that the set I = {e c S : ee ~ = e} and A = {f c S : fof = f} are ~-unipotent and ~-unipotent subbands respectively of S and that, if gO ~ E(SO), then g~ = g~176 for each e e I and fgO = fOgO for each f s A. By using the results, we have: