PROPERTIES OF HYPERIDEALS IN ORDERED SEMIHYPERGROUPS

Abstract

An ordered semihypergroup is a semihypergroup (S; ∘) together with a partial order ≤ on S such that the monotone condition holds, i.e., for all x, y, a ∈ S,  if x ≤ y, then for all u ∈ x ∘ a there exists v ∈ y ∘ a such that u ≤ v, and similarly, for all u' ∈ a ∘ x there exists v' ∈ a ∘ y such that u' ≤ v'.  Indeed, the concept of ordered semihypergroups is a generalization of the concept of ordered semigroups.  In this paper, we study some aspects of hyperideals of ordered semihypergroups.  We give a necessary and suffcient condition of a subset of Cartesian product of two ordered semihypergroups to be a prime hyperideal.  Also, we study right simple element ordered semihypergroups containing right simple elements.