Relationship between ordered semihypergroups and ordered semigroups by using pseudoorder

It is well known that if (S,∘)(S,∘) is a semihypergroup and ρρ is a strongly regular relation on SS, then S/ρS/ρ, the set of equivalence classes, is a semigroup with the binary operation: ρ(x)⊙ρ(y)=ρ(z)ρ(x)⊙ρ(y)=ρ(z) for all z∈x∘yz∈x∘y. Now, let (S,∘,≤)(S,∘,≤) be an ordered semihypergroup. The following question is natural: Is there a strongly regular relation ρρ on SS for which S/ρS/ρ is an ordered semigroup? One of our main aim in this paper is reply to the above question. Then, we study some properties and isomorphisms between them.