n-hypergroups and binary relations

In this paper, we introduce and study the notion of a partial n -hypergroupoid, associated with a binary relation. Some important results concerning Rosenberg partial hypergroupoids, induced by relations, are generalized to the case of n -hypergroupoids. Then, n -hypergroups associated with union, intersection, products of relations and also mutually associative n -hypergroupoids are analyzed. Finally, n -hypergroupoids associated with relations are used in order to study the transitivity of the relation β in n -semihypergroups.