Binary Relations

Abstract

The first connection between a hyperstructure and a binary reIation is implicit in Nieminen [300], who associated a hypergroup with a connected simple graph.

In the same direction, albeit with different hyperoperations associated with graphs, went the papers by Corsini ([74], [79]) and Rosenberg ([326]) and, in the following, by V. Leoreanu and L. Leoreanu ([238]).

Later, Chvalina ([38]) found a correspondence between partially ordered sets and hypergroups. Rosenberg ([326]) generalized Chvalina definition, associating with any binary relation a hypergroupoid.

Rosenberg hypergroup was studied by Corsini ([79]) and then, by Corsini and Leoreanu ([88]), who considered hypergroups associated with union, intersection, product, Cartesian product, direct limit of relations, as we have seen before.

There are stilI open problems on this subject. One of them is to find necessary and sufficient conditions for the hypergroupoids associated with union, intersection, product etc, to be hypergroups. Recently, Spartalis, De SaIvo and Lo Faro have obtained new results on hyperstructures associated with binary relations.