Semilattice strongly regular relations on ordered $n$-ary semihypergroups

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