Multivalued binary relations and Post algebras

This paper deals with a multivalued extension of the concept of a binary relation on a set E. If R is such an r-valued relation on E, far every (x, y)∈E², there exists exactly one i∈{0, 1, ..., r-1} such that the degree of comparability of (x, y) with respect to R is equal to i. The set of all r-valued relations on E is then equipped with an order relation ⩽ and turns out to be a Post algebra of order r. This first part of the paper makes a large use of the results of a previous paper (1999 a). In a second step, we study multivalued product of relations, as well as multivalued reflexivity, symmetry, antisymmetry, and transitivity. It follows for example that, from a multivalued viewpoint, symmetry and antisymmetry turn out to be quite different, since the latter is hereditary while the former is not. We finally investigate some properties of multivalued equivalences, multivalued orders and preorders