On ordered semigroups which are semilattices of left simple semigroups

Abstract It has been proved by Tôru Saitô that a semigroup S is a semilattice of left simple semigroups, that is, it is decomposable into left simple semigroups, if and only if the set of left ideals of S is a semilattice under the multiplication of subsets, and that this is equivalent to say that S is left regular and every left ideal of S is two-sided. Besides, S. Lajos has proved that a semigroup S is left regular and the left ideals of S are two-sided if and only if for any two left ideals L 1, L 2 of S, we have L 1 ∩ L 2 = L 1 L 2. The present paper generalizes these results in case of ordered semigroups. Some additional information concerning the semigroups (without order) are also obtained.