Definability of Relations by Semigroupsof Isotone Transformations

In 1961, L.M. Gluskin proved that a given set with an arbitrary nontrivial quasiorder is determined up to isomorphism or anti-isomorphism by the semigroup of all isotone transformations of , i.e., the transformations of preserving . Subsequently, L.M. Popova proved a similar statement for the semigroup of all partial isotone transformations of ; here the relation does not have to be a quasiorder but can be an arbitrary nontrivial reflexive or antireflexive binary relation on the set . In the present paper, under the same constraints on the relation , we prove that the semigroup of all isotone binary relations (set-valued mappings) of determines up to an isomorphism or anti-isomorphism as well. In addition, for each of the conditions , , and , we enumerate all -ary relations satisfying the given condition.