A further study on the hyperideals of ordered semihypergroups

Abstract

In this paper, we first introduce the concepts of prime, weakly prime and semiprime hyperideals in ordered semihypergroups, and give some characterizations of them.  Furthermore, we consider the extensions of hyperideals in commutative ordered semihypergroups.  As a generalization of the concept of prime hyperideals of ordered semihypergroups, the concept of n-prime hyperideals of ordered semihypergroups is introduced, and related properties are discussed.  In particular, we prove that every (n−1)-prime hyperideal of ordered semihypergroups is n-prime for any positive integer n ≥ 3.  Moreover, we investigate the relationship between n-prime hyperideals and extensions of hyperideals, and prove that a hyperideal I of a commutative ordered semihypergroup is n-prime if and only if any extension of I is (n−1)-prime (n ≥ 3).  Finally, we prove that if I is a semiprime hyperideal of a commutative ordered semihypergroup S, then I is the intersection of all extensions of I.  Especially, if I is also n-prime (n ≥ 3), then I can be expressed as the intersection of all (n−1)-prime hyperideals of S containing it.