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.
$le$-semigroup, left (right) ideal element, ideal element, prime, weakly prime, semiprime, intraregular
Turkish Journal of Mathematics: Vol. 40:
2, Article 7.
Available at: https://journals.tubitak.gov.tr/math/vol40/iss2/7