Definability of Relations by Semigroups of Isotone Transformations

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 reflexive or 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 of the 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.