On $le$-semigroups

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.