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