Hypergroups and binary relations

The paper deals with a binary relation R on a set H, where the Rosenberg partial hypergrougoid H-R is a hypergroup. It Droves that if H-R is a hypergroup, S is an extension of R contained in the transitive closure of R and S subset of 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-RUS is Proven to be a hypergroup. Lastly, a tree T and an iterative sequence of hyperoperations (k) over circle where k = 1, 2, ...) on its verticcs are considered. A bound on the diameter of T is given for each k such that (k) over circle is associative.