Characterizations of (strongly) ordered regular relations on ordered semihypergroups

In this paper, we give the fundamental normal homomorphism theorem of ordered semihypergroups. Moreover, we characterize the (strongly) ordered regular relations on an ordered semihypergroup by means of the ρ-chain, and obtain that the set of all strongly ordered regular relations on an ordered semihypergroup consists a complete lattice. Finally, the subset of an ordered semihypergroup can serve as some ordered regular (ordered semilattice) relation class is determined.