ON LEFT(RIGHT) SEMI-REGULAR po-SEMIGROUPS

Abstract. We give the characterization of left(right) semi-regular po-semigroups and Ve-semigroups. J. Calais([1]) proved that there are semigroups which are not regu-lar, in which the left ideals are idempotent. The problem to describe the class of semigroups in which the left ideals are idempotent is due to S. Lajos([4]). He proved that normal semigroup is regular if and only if every ideal of S is idempotent. N. Kehayopulu([3]) proved that the same results for ordered semigroups and Ve-semigroups are hold. Independently, for the way we work to apply the results of ordered semigroups or of poe(∨e)-semigroups based on ideal elements to semi-groups-without order- we refer to [2]. In this paper, we give a characterization of the left(right) semi-regular po-semigroups and poe-semigroup, respectively. These are the improvements of Kehayopulu’s results(see Corollaries 1 and 3). A po-semigroup(: ordered semigroup) is an ordered set S at the same time a semigroup such that a ≤ b = ⇒ xa ≤ xb and ax ≤ bx for all x ∈ S. A poe-semigroup is a po-semigroup with the greatest element e. A ∨e-semigroup is a poe-semigroup S at same time an upper semilattice satisfying the property a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc for all a, b, c ∈ S. We denote (H] = {x ∈ S|x ≤ h for some h ∈ H} for a subset H of a po-semigroup S. Then we can easily prove the followings; (1) A ⊆ (A] for any A ⊆ S. (2) (A] ⊆ (B] for A ⊆ B ⊆ S. (3) A = (A] for some types of ideal A.