An algebraic approach to binary relations

Abstract

Several attempts were made to assign to a given binary relation a certain binary operation in order to allow an algebraic approach for investigating binary relations. However, the previous attempts by the first two authors were restricted to the case of so-called directed binary relations. In this paper, this restriction is omitted and a general approach is developed. We assign to every binary relation a partial binary operation in such a way that the properties of the relation can be described by properties of the assigned operation. These properties are expressed mostly in terms of existential or strong identities. The partial binary operation can be extended to an everywhere defined binary operation by means of the so-called one-point extension. This enables us to get a genuine algebraic approach similar to that for directed binary relations.