The corresponding results on semigroups (without order) can be also obtained as an application of the results of this paper. The study of poe-semigroups plays an essential role in the theory of fuzzy semigroups and the theory of hypersemigroups. | Turk J Math (2016) 40: 310 – 316 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On le-semigroups Niovi KEHAYOPULU∗ Department of Mathematics, University of Athens, Panepistimiopolis, Athens, Greece Received: • Accepted/Published Online: • Final Version: Abstract: We characterize the idempotent ideal elements of the le -semigroups in terms of semisimple elements and we prove, among others, that the ideal elements of an le -semigroup S are prime (resp. weakly prime) if and only if they form a chain and S is intraregular (resp. semisimple). The corresponding results on semigroups (without order) can be also obtained as an application of the results of this paper. The study of poe -semigroups plays an essential role in the theory of fuzzy semigroups and the theory of hypersemigroups. Key words: le -semigroup, left (right) ideal element, ideal element, prime, weakly prime, semiprime, intraregular 1. Introduction and prerequisites A po-groupoid S is a groupoid under a multiplication “.” and at the same time an ordered set under an order “ ≤ ” such that a ≤ b implies ac ≤ bc and ca ≤ cb for all c ∈ S . If the multiplication of S is associative, then S is called a po-semigroup. A ∨ -groupoid is a groupoid that is also a semilattice under ∨ such that a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc for all a, b, c ∈ S . A ∨ -groupoid that is also a lattice is called an l -groupoid [1,3]. A po -groupoid, ∨ -groupoid, or po -semigroup possessing a greatest element “e” (that is, e ≥ a for any a ∈ S ) is called poe-groupoid, ∨e-groupoid, or poe-semigroup, respectively. Let S be a po -groupoid. An element a of S is called idempotent if a2 = a. An element a of S is called a left ideal element if xa ≤ a for all x ∈ S . It is called a right ideal element if ax ≤ a for all x ∈ S [1]. It is called an ideal element if it is both a left and right ideal element. If S is a .
