Definability of Relations by Semigroups of Isotone Transformations
A. A. Klyushin, Игорь Борисович Кожухов, D. Yu. Manilov, A. V. Reshetnikov · Journal of Applied and Industrial Mathematics · 2024
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.