In this paper, we prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let [Formula: see text] be an NIP field or expansion of a field. Let [Formula: see text] be a definable ring topology on [Formula: see text]. Then [Formula: see text] is a field topology, and [Formula: see text] is locally bounded. If [Formula: see text] has characteristic [Formula: see text] or finite dp-rank, then [Formula: see text] is “generalized [Formula: see text]-henselian” in the sense of Dittman, Walsberg and Ye, meaning that the implicit function theorem holds for polynomials. If [Formula: see text] has finite dp-rank, then [Formula: see text] must be a topology of “finite breadth” (a [Formula: see text]-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian.

We use side condition techniques to introduce a natural stationary set preserving forcing P c c ( λ , μ ) \mathbb {P}^{\mathbf {cc}}(\lambda ,\mu ) that (under N S ω 1 \mathsf {NS}_{\omega _1} precipitous + existence of H θ # H_{\theta }^\# for a sufficiently large regular θ \theta ) increases the second uniform indiscernible u 2 \mathbf {u}_2 beyond some given ordinal λ \lambda . The forcing P c c \mathbb {P}^{\mathbf {cc}} shares this property with forcings defined by Claverie and Schindler [J. Symbolic Logic 74 (2009), pp. 187–200] and Ketchersid, Larson, and Zapletal [J. Symbolic Logic 72 (2007), pp. 1372–1378]. We show that P c c \mathbb {P}^{\mathbf {cc}} is in addition strongly ssp. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games G M c p ( X ) \mathbf {G}^{\mathbf {cp}}_M(X) and the catching-capturing games G M c c ( X ) \mathbf {G}^{\mathbf {cc}}_M(X) . In particular, these games are used to isolate a special family of countable elementary submodels M ≺ H μ M \prec H_\mu that occur as side conditions in P c c \mathbb {P}^{\mathbf {cc}} and thus allow to control the forcing in a strong way.

Abstract The author previously defined the divisible completion of the solvable Baumslag–Solitar group, described the groups elementarily equivalent to it, constructed an axiomatization of the corresponding theory, and proved its decidability. In the present work, elementary submodels of models of this theory are described, and $ \omega $ -stability is established.

In this article, we study the expansion of a structure by adding a new predicate that is not definable by any formula in the original language. To consider an externally definable expansion, we define the extension of a model in both essential and non-essential case. Such expansions can lead to significant changes in the properties of the resulting structure. We focus on the case of externally definable expansions, where the new relation is given by the intersection of a formula defined in an elementary extension with the original structure. The concept of a uniformly externally definable expansion was first introduced by Macpherson, Marker, and Steinhorn in the context of expansions by cuts in submodels of o-minimal structures over the real numbers. Subsequently, Baizhanov demonstrated that expanding a model of a weakly o-minimal theory by a family of convex sets preserves both weak o-minimality and uniform external definability. We establish conditions for external expansions under which the key properties of the original structure are preserved.

Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra). 9 pages. Published in the journal of Groups, Complexity, Cryptology

In this note we prove several theorems that are related to some results and problems from [6] . We answer two of the main questions that were raised in [6] . First we give a ZFC example of a Hausdorff space in C ( ω 1 ) that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every regular space in C ( ω 1 ) has countable net weight. In the last section we prove in ZFC the following two statements: (i) If S ⊂ ω 1 is stationary then for any regular topology on S of uncountable weight S has a non-stationary subset that has uncountable weight as well. (ii) For any topology on ω 1 , if all final segments of ω 1 have uncountable weight then ω 1 has a non-stationary subset of uncountable weight. In contrast to this, it was shown in [6] that the analogous statements for net weight are not provable in ZFC. It is remarkable that all our proofs of the above results make essential use of elementary submodels.

We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors L(Fn) share a common first-order theory? We establish a trichotomy of possibilities for their common first-order fundamental group, as well as several possible avenues for establishing a dichotomy in direct analog to the free group factor alternative of Dykema and Radulescu. We also show that the ∀∃-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of ∀∃-theories. We conclude with some observations on related problems.

Abstract Working within the context of countable, superstable theories, we give many equivalents of a theory having NOTOP. In particular, NOTOP is equivalent to V-DI, the assertion that any type V-dominated by an independent triple is isolated over the triple. If T has NOTOP, then every model N is atomic over an independent tree of countable, elementary substructures, and hence is determined up to back-and-forth equivalence over such a tree. We also verify Shelah’s assertion from Chapter XII of [9] that NOTOP implies PMOP (without using NDOP).

Ehrenfeucht-Fraïssé games provide means to characterize elementary equivalence for first-order logic, and by standard translation also for modal logics. We propose a novel generalization of Ehrenfeucht-Fraïssé games to hybrid-dynamic logics which is direct and fully modular: parameterized by the features of the hybrid language we wish to include, for instance, the modal and hybrid language operators as well as first-order existential quantification. We use these games to establish a new modular Fraïssé-Hintikka theorem for hybrid-dynamic propositional logic and its various fragments. We study the relationship between countable game equivalence (determined by countable Ehrenfeucht-Fraïssé games) and bisimulation (determined by countable back-and-forth systems). In general, the former turns out to be weaker than the latter, but under certain conditions on the language, the two coincide. As a corollary we obtain an analogue of the Hennessy-Milner theorem. We also prove that for reachable image-finite Kripke structures elementary equivalence implies isomorphism.

Abstract Ancel, Dobrowolski and Grabowski (Studia Math. 109 (1994): 277–290) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set‐theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.

Recall the Rabin-Keisler theorem which gives a lower bound κω for the size of proper elementary extensions of complete structures of size κ, provided that κ is an infinite cardinal below the first measurable cardinal. We survey – and at places clarify and extend – some facts which connect the Rabin-Keisler theorem, sizes of ultrapowers, combinatorial properties of ultrafilters, and large cardinals.

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of F p ( t ) h {\mathbb {F}}_p(t)^h is an elementary substructure of the perfect hull of F p ( ( t ) ) {\mathbb {F}}_p(\!(t)\!) .

Abstract The lattice problem for models of Peano Arithmetic ( $\mathsf {PA}$ ) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf {PA}$ , or, in greater generality, for a given model $\mathcal {M}$ , which lattices can be represented as interstructure lattices of elementary submodels $\mathcal {K}$ of an elementary extension $\mathcal {N}$ such that $\mathcal {M}\preccurlyeq \mathcal {K}\preccurlyeq \mathcal {N}$ . The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.

Abstract We prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals of tracial factors implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś’s theorem and countable saturation, to this more general setting.

To efficiently compress the sign information of images, we address a sign retrieval problem for the block-wise discrete cosine transformation (DCT): reconstruction of the signs of DCT coefficients from their amplitudes. To this end, we propose a fast sign retrieval method on the basis of binary classification machine learning. We first introduce 3D representations of the amplitudes and signs, where we pack amplitudes/signs belonging to the same frequency band into a 2D slice, referred to as the sub-band block. We then retrieve the signs from the 3D amplitudes via binary classification, where each sign is regarded as a binary label. We implement a binary classification algorithm using convolutional neural networks, which are advantageous for efficiently extracting features in the 3D amplitudes. Experimental results demonstrate that our method achieves accurate sign retrieval with an overwhelmingly low computation cost.

Abstract We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the set $D'$ comprised of differences between successive elements. In particular, if the burden of the structure is at most n, then the result of applying the operation $D \mapsto D'\ n$ times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden $2$ , we show that any definable unary discrete set must be definable in some elementary extension of the structure $\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).

Tensoring with type I algebras preserves elementary equivalence in the category of tracial von Neumann algebras. The proof involves a novel and general Feferman-Vaught-type theorem for direct integrals of metric structures.

Abstract The main motivation of this paper is the study of first-order model theoretic properties of structures having their roots in modal logic. We will focus on the connections between ultrafilter extensions and ultrapowers. We show that certain structures (called bounded graphs) are elementary substructures of their ultrafilter extensions, moreover their modal logics coincide.

Elementary Equivalence of Stable Linear Groups Over Fields of Characteristic 2

In this paper we prove the criterion of elementary equivalence of linear groups over graded rings with finite number of centralidempotents from the 0-component, when grading is partially included in the group language. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.