?
Замкнутые классы инфинитарных функций и их приложения в теории ультрафильтров
С. 102–112.
The Galois theory for closed classes of infinitary functions and some of its applications in the theory of ultrafilters are considered.
Polyakov N. L., Saveliev D. I., Russian Mathematical Surveys 2026 Vol. 81 No. 1 P. 205–206
We solve Problem 61 from Hart and van Mill’s list on whether every finite partial order is embeddable in the Rudin–Keisler order on (types of) ultrafilters over $\omega$. ...
Added: January 31, 2026
Polyakov N. L., Шамолин М. В., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2025 Т. 525 С. 135–143
In this paper, Galois theory is developed for closed sets of functions of any ordinal arity. The classical theorem on Galois-closed classes of functions and sets of predicates on finite sets is transferred to the general case. ...
Added: December 12, 2025
Polyakov N. L., Saveliev D., Working papers by Cornell University. Series math "arxiv.org" 2025 P. 1–13
A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on (2024), asks whether every finite partial order is embeddable in the Rudin--Keisler order on (types of) ultrafilters over a countable set. Although the positive answer, even for all countable partial orders, was proved under CH in Blass' thesis ...
Added: November 25, 2025
Polyakov N. L., В кн.: Algebra and model theory 2025. Volume 16.: ., 2025. С. 134–139.
In [1], the concept of a skew limit ultrapower of ordinal rank of an arbitrary model $\mathfrak M$ with respect to an ultrafilter $\mathfrak u$ was introduced for a model-theoretic characterization of some natural preorders on the set of ultrafilters on the set $\beta\omega$. We will show that this concept admits broad generalizations that can ...
Added: November 24, 2025
N. L. Poliakov, Saveliev D., Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series 2025 Vol. 151 No. 2 P. 6–11
Generalizing of the Rudin--Keisler preorder, we introduce relations $R_\alpha$ (and $R_{<\alpha}$) on the set $\scc\omega$ of ultrafilters on~$\omega$. They form an ordinal sequence of length~$\omega_1$ which is strictly increasing by inclusion and lies between the Rudin--Keisler preorder and the Comfort preorder. We show that the composition of these relations is expressed ...
Added: July 16, 2025
Polyakov N. L., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2025 Т. 522 № 1 С. 40–49
The paper describes a new method for constructing graphs without triangles and with an arbitrarily large chromatic number. The properties of various types of ultrafilter extensions of functions and predicates are used to justify the method. ...
Added: June 3, 2025
Polyakov N. L., В кн.: Международная конференция МАЛЬЦЕВСКИЕ ЧТЕНИЯ 13–17 ноября 2023 г. Тезисы докладов.: [б.и.], 2023. С. 104–104.
Added: November 30, 2024
N. L. Polyakov, , in: Model Theory and Algebra 2024.: -, 2024. P. 87–92.
We describe the Rudin-Keisler preorder on lower cones with respect to the Comfort preorder of Ramsey ultrafilters ...
Added: November 18, 2024
Мир Н. А., Polyakov N. L., Чебышевский сборник 2024 Т. 25 № 3 С. 396–407
The paper gives a short proof of the canonical Ramsey theorem of Erd˝os and Rado using ultrafilter theory. ...
Added: November 18, 2024
Н. Л. Поляков, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2023 Т. 513 С. 76–87
We give a characterization of Ramsey ultrafilters on \omega in terms of functions f:\omega^n\to\omega and their ultrafilter extensions. To do this, we prove that for any partition P of [\omega]^n there is a finite partition Q of [\omega]^{2n} such that any set X\subseteq \omega that is homogeneous for Q is a finite union of sets that ...
Added: November 23, 2023
Esterov A. I., Lang L., Selecta Mathematica, New Series 2022 Vol. 28 No. 2 Article 22
We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry.
As a model example, consider the problem of permuting the roots of a complex polynomial
f(x) = c0 + c1 x^d1 +. . .+ ck x^dk
by varying its coefficients. ...
Added: June 22, 2022