A further study on ordered regular equivalence relations in ordered semihypergroups

In this paper, we study the ordered regular equivalence relations on ordered semihypergroups in detail. To begin with, we introduce the concept of weak pseudoorders on an ordered semihypergroup, and investigate several related properties. In particular, we construct an ordered regular equivalence relation on an ordered semihypergroup by a weak pseudoorder. As an application of the above result, we completely solve the open problem on ordered semihypergroups introduced in [B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseuoorders, European J. Combinatorics 44 (2015), 208–217]. Furthermore, we establish the relationships between ordered regular equivalence relations and weak pseudoorders on an ordered semihypergroup, and give some homomorphism theorems of ordered semihypergroups, which are generalizations of similar results in ordered semigroups