Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H  . In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρρ and σσ on H   we investigate when the hypergroups associated with the relations ρ∩σρ∩σ, ρ∪σρ∪σ and ρσρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.