Semigroups which contain magnifying elements are factorizable

A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma).

In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.