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Ternary semihypergroups are algebraic structures with one associative hyperoperation. The main propose of this article is to study binary relations on ternary semihypergroups and study some basic properties of compatible relations on them. In particular, we analyze the ternary hypergroup associated with a binary relation.

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The paper deals with a binary relation R on a set H , where the Rosenberg partial hypergroupoid H R is a hypergroup. It proves that if H R is a hypergroup, S is an extension of R contained in the transitive closure of R and S ⊂ S 2 , then H S is also a hypergroup. Corollaries for various extensions of R, the union, intersection and product constructions being employed, are then proved. If H R and H S are mutually associative hypergroups then H R∪S is proven to be a hypergroup. Lastly, a tree T and an iterative sequence of hyperoperations • k where k = 1, 2, . . .) on its vertices are considered. A bound on the diameter of T is given for each k such that • k is associative.

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