Ordered Semigroups in which the Left Ideals are Intra-Regular Semigroups

It is well known that intra-regular semigroups (ordered semigroups) play an essential role in studying the structure of semigroups (ordered semigroups). In this paper we present a structure theorem referring to the decomposition of ordered semigroups into left strongly simple components, that is, into components, which are both simple and left quasi-regular. We prove, among others, that if an ordered semigroup S is a union of its left strongly simple subsemigroups, then every left ideal of S is an intra-regular subsemigroup of S. ”Conversely” if every left ideal of S is an intra-regular subsemigroup of S, then S is a complete semilattice of left strongly simple semigroups. As a consequence, an ordered semigroup S is a semilattice of left strongly simple semigroups if and only if it is a complete semilattice of left strongly simple semigroups. We also characterize the chains of left strongly simple ordered semigroups.