Summary: 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.