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.