Geodesic
Definability of relations by semigroups of isotone transformations
Diskretnyj analiz i issledovanie operacij, Tome 31 (2024) no. 1, pp. 19-34 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: semigroup of binary relations
Mots-clés : isotone transformation. 
@article{DA_2024_31_1_a1,
     author = {A. A. Klyushin and I. B. Kozhukhov and D. Yu. Manilov and A. V. Reshetnikov},
     title = {Definability of relations by semigroups of~isotone transformations},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {19--34},
     year = {2024},
     volume = {31},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2024_31_1_a1/}
}
TY  - JOUR
AU  - A. A. Klyushin
AU  - I. B. Kozhukhov
AU  - D. Yu. Manilov
AU  - A. V. Reshetnikov
TI  - Definability of relations by semigroups of isotone transformations
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2024
SP  - 19
EP  - 34
VL  - 31
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DA_2024_31_1_a1/
LA  - ru
ID  - DA_2024_31_1_a1
ER  - 
%0 Journal Article
%A A. A. Klyushin
%A I. B. Kozhukhov
%A D. Yu. Manilov
%A A. V. Reshetnikov
%T Definability of relations by semigroups of isotone transformations
%J Diskretnyj analiz i issledovanie operacij
%D 2024
%P 19-34
%V 31
%N 1
%U http://geodesic.mathdoc.fr/item/DA_2024_31_1_a1/
%G ru
%F DA_2024_31_1_a1
A. A. Klyushin; I. B. Kozhukhov; D. Yu. Manilov; A. V. Reshetnikov. Definability of relations by semigroups of isotone transformations. Diskretnyj analiz i issledovanie operacij, Tome 31 (2024) no. 1, pp. 19-34. http://geodesic.mathdoc.fr/item/DA_2024_31_1_a1/

[1] L. M. Gluskin, “Semigroups of isotone transformations”, Usp. Mat. Nauk, 16:5 (1961), 157–162 (Russian) | MR | Zbl

[2] L. M. Popova, “Semigroups of partial endomorphisms of a set with a relation”, Sib. Mat. Zh., 4:2 (1963), 309–317 (Russian) | Zbl

[3] I. B. Kozhukhov and V. A. Yaroshevich, “Transformation semigroups preserving a binary relation”, J. Math. Sci., 164:2 (2010), 240–244 | DOI | MR | Zbl

[4] Molchanov V. A., “Semigroups of mappings on graphs”, Semigroup Forum, 27 (1983), 155–199 | DOI | MR | Zbl

[5] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, v. 1, AMS, Providence, 1961 | MR | Zbl

[6] Yu. L. Ershov and E. A. Palyutin, Mathematical Logic, Nauka, M., 1987 (Russian) | MR

[7] B. I. Plotkin, Automorphism Groups of Algebraic Systems, Nauka, M., 1966 (Russian)

[8] V. V. Vagner, “Relation theory and algebra of partial mappings”, Theory of Semigroups and Its Applications, Izd. Saratov. Univ., Saratov, 1965, 3–178 (Russian)