About the fundamental relations defined on the hypergroupoids associated with binary relations

This paper deals with connections between hypergroupoids and nn-ary relations. First we prove that the study of the reduced hypergroupoids associated with nn-ary relations can be linked to the simpler case of binary relations. On the basis of the properties of some fundamental relations defined on a hypergroupoid, we determine necessary and sufficient conditions for two elements in a hypergroupoid associated with a binary relation to be operationally equivalent or inseparable; moreover we characterize the reduced hypergroupoids in some special cases.