On Prime, Weakly Prime Ideals in Semigroups

Semigroups in which the ideals are prime, weakly Prime have been considered by G. Szasz in [4]. G. Szasz has shown that the ideals of a semigroup P are Prime if and only if P is intra-regular [2] and the ideals of P form a chain, the ideals of P are weakly. Prime if and only if they have idempotency and form a chain. He also proved that an ideal of a semi-group P is Prime if and only if it is both semiprime and weakly prime, and that in commutative semigroups the prime and weakly prime ideals coincide. On the other hand, the concept of intra-regular Po-semi-group has been introduced in [3] and extends the concept of intra-regular semigroups given by A.H. Clifford and G.B. Preston [3] in case of ordered semigroups. Here we consider semigroups which do not necessarily have greatest idempotent element, we introduce the concepts of ideals, and intra-regular semigroup.