 -hypergroups and  -semihypergroups associated to binary relations

The concept of  -semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of  -semigroups. In this paper, we study the concept of semiprime ideals in a  -semihypergroup and prove some results. Also, we introduce the notion of  -hypergroups and closed  -subhypergroups. Finally, we study the concept of  -semihypergroups associated to binary relations and give necessary and sufficient conditions on a set of binary relations  on a non-empty set S such that S becomes a  -semihypergroup or a  -hypergroup.

Hypergroup; semihypergroup;  -semigroup;  -semihypergroup; binary relation