Semilattice strongly regular relations on ordered n-ary semihypergroups

In this paper, we introduce the concept of j-hyperfilters, for all positive integers 1 <= j <= n and n >= 2, on (ordered) n-ary semihypergroups and establish the relationships between j-hyperfilters and completely prime j-hyperideals of (ordered) n-ary semihypergroups. Moreover, we investigate the properties of the relation N, which is generated by the same principal hyperfilters, on (ordered) n-ary semihypergroups. As we have known from [21] that the relation N is the least semilattice congruence on semihypergroups, we illustrate by counterexample that the similar result is not necessarily true on nary semihypergroups where n >= 3. However, we provide a sufficient condition that makes the previous conclusion true on n-ary semihypergroups and ordered n-ary semihypergroups where n >= 3. Finally, we study the decomposition of prime hyperideals and completely prime hyperideals by means of their N-classes. As an application of the results, a related problem posed by Tang and Davvaz in [31] is solved.