-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.