The Semigroups of All Binary Relations on Set

The article considers the properties of the semigroup R(X) of all binary relations on a non-empty set X with the operation of composition of binary relations. The authors pro-vide a proof that all information about binary relations on X is expressed in terms of the semigroup R(X). They describe isomorphisms between the semigroups R(X) and R(Y) for arbitrary sets X and Y. The paper also gives an abstract characterization of the semigroups R(X).