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 $T_\rho(X)=T(X)$, $P_\rho(X)=P(X)$, $B_\rho(X)=B(X),$ we enumerate all $n$-ary relations $\rho$ satisfying the given condition. Bibliogr. 8.