An investigation on hyper S-posets over ordered semihypergroups

In this paper, we define and study the hyper -posets over an ordered semihypergroup in detail. We introduce the hyper version of a pseudoorder in a hyper -poset, and give some related properties. In particular, we characterize the structure of factor hyper -posets by pseudoorders. Furthermore, we introduce the concepts of order-congruences and strong order-congruences on a hyper -poset , and obtain the relationship between strong order-congruences and pseudoorders on . We also characterize the (strong) order-congruences by the -chains, where is a (strong) congruence on . Moreover, we give a method of constructing order-congruences, and prove that every hyper -subposet of a hyper -poset is a congruence class of one order-congruence on if and only if is convex. In the sequel, we give some homomorphism theorems of hyper -posets, which are generalizations of similar results in -posets and ordered semigroups.