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Denis I. Saveliev

Publications and source records attributed to Denis I. Saveliev.

15 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.

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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.

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arXiv2024

Hindman's finite sums theorem and its application to topologizations of algebras

The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups. The second, main part of the paper is devoted to the topologizability problem of a wide class of algebraic structures called polyrings; this class includes Abelian groups, rings, modules, algebras over a ring, differential rings, and others. We show that the Zariski topology of such an algebra is always non-discrete. Actually, a much stronger fact holds: if $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$. The proof essentially uses a multidimensional version of Hindman's finite sums theorem. The third part of the paper lists several problems concerning topologization of various algebraic structures, their Zariski topologies, and related questions.

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arXiv2023

A note on restriction rules

We consider a certain class of infinitary rules of inference, called here restriction rules, using of which allows us to deduce complete theories of given models. The first instance of such rules was the $ω$-rule introduced by Hilbert, and generalizations of the $ω$-rule were first considered by Henkin. Later on Barwise showed that within countable languages, for any countable model $\mathfrak M$, first-order logic expanded by the corresponding $\mathfrak M$-rule deduces from the diagram of $\mathfrak M$ all formulas that are true in all models that include $\mathfrak M$, in particular, it deduces the (relativized) complete theory of $\mathfrak M$. We show that, if the aim is only deducing the complete theory of a given model, these countability assumptions can be omitted. Moreover, similar facts hold for infinitary and higher-order logics, even if these logics are highly incomplete. Finally, we show that Barwise's theorem in its stronger form, for vocabularies of arbitrary cardinality, holds for infinitary logics $\mathscr L_{κ,λ}$ whenever $κ$ is supercompact. The note also contains brief historical remarks.

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arXiv2021

On reduction and separation of projective sets in Tychonoff spaces

We show that for every Tychonoff space $X$ and Hausdorff operation $\mathbfΦ$, the class $\mathbfΦ(\mathscr Z,X)$ generated from zero sets in $X$ by $\mathbfΦ$ has the reduction or separation property if the corresponding class $\mathbfΦ(\mathscr F,\mathbb R)$ of sets of reals has the same property. In particular, under Projective Determinacy, these properties of such projective sets in $X$ have the same pattern as the First Periodicity Theorem states for projective sets of reals: the classes $\mathbfΣ^{1}_{2n}(\mathscr Z,X)$ and $\mathbfΠ^{1}_{2n+1}(\mathscr Z,X)$ have the reduction property while $\mathbfΠ^{1}_{2n}(\mathscr Z,X)$ and $\mathbfΣ^{1}_{2n+1}(\mathscr Z,X)$ have the separation property.

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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.

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arXiv2020

On two types of ultrafilter extensions of binary relations

There exist two distinct types of ultrafilter extensions of binary relations, one discovered in universal algebra and modal logic, and another, in model theory and algebra of ultrafilters. We show that the extension of the latter type is properly included in the extension of the former type, and describe their interaction with the relation algebra operations. Then we provide topological characterizations of both extensions and show that the larger extension continuously maps the space of ultrafilters into the space of filters endowed with the Vietoris topology.

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arXiv2019

On modal logics of model-theoretic relations

Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal operator is interpreted via $\mathcal R$. We discuss how modal theories of $\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

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arXiv2019

On first-order expressibility of satisfiability in submodels

Let $κ,λ$ be regular cardinals, $λ\leκ$, let $φ$ be a sentence of the language $\mathcal L_{κ,λ}$ in a given signature, and let $\vartheta(φ)$ express the fact that $φ$ holds in a submodel, i.e., any model $\mathfrak A$ in the signature satisfies $\vartheta(φ)$ if and only if some submodel $\mathfrak B$ of $\mathfrak A$ satisfies $φ$. It was shown in [1] that, whenever $φ$ is in $\mathcal L_{κ,ω}$ in the signature having less than $κ$ functional symbols (and arbitrarily many predicate symbols), then $\vartheta(φ)$ is equivalent to a monadic existential sentence in the second-order language $\mathcal L^{2}_{κ,ω}$, and that for any signature having at least one binary predicate symbol there exists $φ$ in $\mathcal L_{ω,ω}$ such that $\vartheta(φ)$ is not equivalent to any (first-order) sentence in $\mathcal L_{\infty,ω}$. Nevertheless, in certain cases $\vartheta(φ)$ are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when $\vartheta(φ)$ is in $\mathcal L_{κ,κ}$ and $κ$ is $ω$ or a certain large cardinal.

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arXiv2013

Ultrafilter extensions of linear orders

It was recently shown that arbitrary first-order models canonically extend to models (of the same language) consisting of ultrafilters. The main precursor of this construction was the extension of semigroups to semigroups of ultrafilters, a technique allowing to obtain significant results in algebra and dynamics. Here we consider another particular case where the models are linearly ordered sets. We explicitly calculate the extensions of a given linear order and the corresponding operations of minimum and maximum on a set. We show that the extended relation is not more an order however is close to the natural linear ordering of nonempty half-cuts of the set and that the two extended operations define a skew lattice structure on the set of ultrafilters.

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arXiv2010

Common idempotents in compact left topological left semirings

A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.

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arXiv2007

Choice and Regularity: Common Consequences in Logic

It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating their "intersection", a principle (called here Best-Foundedness) which is consistent with the negations of both axioms but implies all these consequences. Then we study relationships between these consequences (and near principles) in detail. Finally, we consider some arguments related to truth of various principles in set theory, especially arguments concerning the interpretability strength.

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arXiv2007

A Note on Singular Cardinals in Set Theory Without Choice

We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.

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arXiv2006

A game on the universe of sets

In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are winning, i.e. have a winning strategy for a player, form a natural hierarchy with levels indexed by ordinals. We show that the class of hereditarily winning sets is an inner model containing all well-founded sets, and that all four possible relationships between the universe, the class of hereditarily winning sets, and the class of well-founded sets are consistent. We describe classes of ordinals for which it is consistent that winning sets without minimal elements are exactly in the levels indexed by ordinals of this class. For consistency results, we propose a new method for getting non-well-founded models. Finally, we establish a probability result by showing that on hereditarily finite well-founded sets the first player wins almost always.

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