Applications of Strongly Transitive Geometric Spaces to $n$-Ary Hypergroups

Abstract

$n$-ary hypergroups are a generalization of Dörnte $n$-ary groups and a generalization of hypergroups in the sense of Marty. In this paper, we investigate some properties of $n$-ary hypergroups and (commutative) fundamental relations. We determine two families $P(H)$ and $P_{\sigma }(H)$ of subsets of an $n$-ary hypergroup $H$ such that two geometric spaces $(H,P(H))$ and $(H,P_{\sigma }(H))$ are strongly transitive. We prove that in every $n$-ary hypergroup, the fundamental relation $\beta$ and the commutative fundamental relation $\gamma$ are strongly compatible equivalence relations.