On prime, weakly prime ideals in ordered semigroups

On ordered hypersemigroups with idempotent ideals, prime and weakly prime ideals

“…Let a ∈ H. By hypothesis, we have I(a) = (I(a) * I(a)]. In the implication (4) ⇒ (5) of Lemma 2 in [4], we replace the multiplication "·" by " * ", the proof follows. This is the notion of an intra-regular ordered semigroup introduced by Kehayopulu in [5]: An ordered semigroup (S, ·, ≤) is called intra-regular if for every a ∈ S there exist x, y ∈ S such that a ≤ xa 2 y, that is if a ∈ (Sa 2 S] for every a ∈ S, equivalently if A ⊆ (SA 2 S] for every A ⊆ S. This concept can be naturally transferred to ordered hypersemigroups in the following definition.…”

“…In our paper in Semigroup Forum 44 (1992) 341-346 [4] we characterized the ordered semigroup S in which the ideals are idempotent in terms of the ideals of S and we proved that this type of ordered semigroups are the semisimple ordered semigroups. We also proved that the ideals of an ordered semigroup S are weakly prime if and only if they are idempotent and they form a chain.…”

On quasi-prime, weakly quasi-prime left ideals in ordered-Γ-semigroups

“…It can be easily verified that L is not weakly prime, quasi-prime and weakly 4 are weakly prime, quasi-prime and weakly quasi-prime. If we define a relation ≤ on M as follows: …”

Ordered semigroups whose elements are separated by prime ideals

“…semigroup) S is called prime [1,3] if the complement S\T of T to S is either empty or it is a subsemigroup of S (that is either S\T = ∅ or S\T = ∅ and a, b / ∈ T implies ab / ∈ T ). Equivalent definitions are the following three definitions:…”

“…For A = {a} we write I(a) instead of I({a}). We have I(a) = (a∪aS ∪Sa∪SaS] [3]. S is called intra-regular [6,7] if for every a ∈ S there exist x, y ∈ S such that a ≤ xa 2 y.…”