Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.