Weakly prime and prime fuzzy ideals in ordered semigroups

Our aim is to promote research and the development of fuzzy technology by studying the fuzzy ordered semigroups. The goal is to explain new methodological developments in fuzzy ordered semigroups which will be of growing importance in the future. Intra-regular ordered semigroups play an important role in studying the structure, especially the decomposition of ordered semigroups. In this paper we prove that the fuzzy ideals of an ordered semigroup $S$ are weakly prime if and only if they are idempotent and they form a chain, and that they are prime if and only if $S$ is intra-regular and the fuzzy ideals of $S$ form a chain. Moreover we show that a fuzzy ideal of an ordered semigroup is prime if and only if it is both semiprime and weakly prime and that in commutative ordered semigroups the prime and weakly prime fuzzy ideals coincide. Our results extend the corresponding results on semigroups (without order) given by G. Szász in [15] in case of ordered semigroups using fuzzy sets.