arXiv subjects

Nikolai L. Poliakov

Publications and source records attributed to Nikolai L. Poliakov.

4 recordsLinked to original sources

arXiv2026

On embedding of partially ordered sets in $(βω,\le_{RK})$

A natural question, which appeared as Problem 61 in Hart and van Mill's list of open problems on $βω$ (2024), asks whether every finite partial order is embeddable in the Rudin--Keisler order on the set of ultrafilters over a countable set. Although the positive answer, even for all countable partial orders, was proved under CH in Blass' thesis (1970), the question in ZFC alone remained completely open. We show that, in ZFC, it is possible not only to answer in the positive for all countable orders, but, moreover, to construct embeddings of the ordered by inclusion lattices of finite subsets of a set of cardinality $2^{\mathfrak c}$, and of countable subsets of a set of cardinality $\aleph_1$, into the set of ultrafilters with any relation lying between the Rudin--Keisler and Comfort orders.

math.GN↗

arXiv2026

On skew ultralimits and their applications in ultrafilter theory

We define a special version of the ultralimit, called the skewed ultralimit. Using this tool, we show that the set of ultrafilter types in the $C$-equivalence class of a Ramsey ultrafilter $\mathfrak u\in βω$ with the Rudin-Keisler order is isomorphic to the ultrapower of $(ω, \leq)$.

math.LO↗

arXiv2026

More on expressibility of satisfiability in submodels and extensions

We study expressibility in infinitary languages of the modal operators associated with satisfiability of sentences of these languages in submodels and extensions of models. We give a syntactic criterion for expressibility in finitary predicate languages, show that in many cases infinitary languages are closed under the operator associated with submodels, and that this is so in any language with a purely monadic signature. Finally, we prove that in finitary or strongly compact languages, the operator associated with extensions, though can be inexpressible by a single sentence, is always expressible by a universal theory, in striking contrast with the submodel case.

math.LO↗

arXiv2021

On ultrafilter extensions of first-order models and ultrafilter interpretations

There exist two known canonical types of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification; the ultrafilter extensions generalize this fact to discrete spaces endowed with an arbitrary first-order structure. Results of such kind are referred to as extension theorems. We offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. Then we propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We establish necessary and sufficient conditions under which ultrafilter models coincide with the canonical ultrafilter extensions of some ordinary models, and obtain their topological characterization. Further we propose even a wider concept of ultrafilter models together with their semantics based on limits of ultrafilters, and show that the new concept absorbs the former one as well as the ordinary concept of first-order models. We establish necessary and sufficient conditions under which ultrafilter models in the wide sense coincide with those in the narrow sense or with the ultrafilter extensions of some ordinary models. Finally we prove extension theorems for ultrafilter models.

math.LO↗