A novel investigation on fuzzy hyperideals in ordered $$*$$-semihypergroups

Abstract

In this paper, we study the ordered $$*$$ -semihypergroups in terms of fuzzy subsets in detail and define a unary operation $$\star $$ on the set of all the fuzzy subsets of an ordered $$*$$ -semihypergroup. To begin with, we define and study the fuzzy hyperideals of an ordered $$*$$ -semihypergroup. In particular, we investigate the properties of fuzzy hyperideals generated by ordered fuzzy points of an ordered $$*$$ -semihypergroup. Furthermore, we introduce the concepts of prime, weakly prime and semiprime fuzzy hyperideals of ordered $$*$$ -semihypergroups. Especially, the relationships among these three types of fuzzy hyperideals are established. In the sequel, we give some characterizations of intra-regular ordered $$*$$ -semihypergroups and semisimple ordered $$*$$ -semihypergroups in terms of fuzzy hyperideals. Especially, we prove that an ordered $$*$$ -semihypergroup S is semisimple if and only if every fuzzy hyperideal of S can be expressed as the intersection of all weakly prime fuzzy hyperideals of S containing it.