A construction of weakly inverse semigroups

Let S (o) be an inverse semigroup with semilattice biordered set E (o) of idempotents and E a weakly inverse biordered set with a subsemilattice E (P) = {e a E vertical bar is not an element of f a E, S(f, e) aS dagger omega(e)} isomorphic to E (o) by theta: E (P) -> E (o). In this paper, it is proved that if is not an element of f, g a E, f a center dot g a(1) f (o) theta D (SA degrees) g (o) theta and there exists a mapping phi from E (P) into the symmetric weakly inverse semigroup P R (Ea(a)S (o)) satisfying six appropriate conditions, then a weakly inverse semigroup I pound can be constructed in P R (S (o)), called the weakly inverse hull of a weakly inverse system (S (o),E, theta, phi) with I(gS) a parts per thousand... S (o), E(I ) pound a parts per thousand integral E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.