Recent zbMATH articles in MSC 03Chttps://zbmath.org/atom/cc/03C2023-11-13T18:48:18.785376ZUnknown authorWerkzeugMini-workshop: Topological and differential expansions of o-minimal structures. Abstracts from the mini-workshop held November 27 -- December 3, 2022https://zbmath.org/1521.000122023-11-13T18:48:18.785376ZSummary: The workshop brought together researchers with expertise in areas of mathematics where model theory has had interesting applications. The areas of expertise spanned from expansions of o-minimal structures preserving tame geometric properties to expansions of specified fields by classical operators that preserve neo-stability properties. There were presentations and discussions on recent developments in definable groups and decompositions in relatively tame setups, the interplay of different notions of dimension and closure operators, and applications of the model theory of differential fields to diophantine geometry.Finite satisfiability for two-variable, first-order logic with one transitive relation is decidablehttps://zbmath.org/1521.030192023-11-13T18:48:18.785376Z"Pratt-Hartmann, Ian"https://zbmath.org/authors/?q=ai:pratt-hartmann.ianSummary: We consider two-variable, first-order logic in which a single distinguished predicate is required to be interpreted as a transitive relation. We show that the finite satisfiability problem for this logic is decidable in triply exponential non-deterministic time. Complexity falls to doubly exponential non-deterministic time if the transitive relation is constrained to be a partial order.Orders on computable ringshttps://zbmath.org/1521.030222023-11-13T18:48:18.785376Z"Wu, Huishan"https://zbmath.org/authors/?q=ai:wu.huishanSummary: The Artin-Schreier theorem says that every formally real field has orders. \textit{H. M. Friedman} et al. showed in [Ann. Pure Appl. Logic 25, 141--181 (1983; Zbl 0575.03038); addendum ibid. 28, 319--320 (1985; Zbl 0575.03039)] that the Artin-Schreier theorem is equivalent to \(\mathsf{WKL}_0\) over \(\mathsf{RCA}_0\). We first prove that the generalization of the Artin-Schreier theorem to noncommutative rings is equivalent to \(\mathsf{WKL}_0\) over \(\mathsf{RCA}_0\). In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre-orders to orders, where Zorn's lemma is used. We then prove that ``pre-orders on rings not necessarily commutative extend to orders'' is equivalent to \(\mathsf{WKL}_0\).The Lyndon property and uniform interpolation over the Grzegorczyk logichttps://zbmath.org/1521.030342023-11-13T18:48:18.785376Z"Maksimova, L. L."https://zbmath.org/authors/?q=ai:maksimova.larisa-lSummary: We consider versions of the interpolation property stronger than the Craig interpolation property and prove the Lyndon interpolation property for the Grzegorczyk logic and some of its extensions. We also establish the Lyndon interpolation property for most extensions of the intuitionistic logic with Craig interpolation property. For all modal logics over the Grzegorczyk logic as well as for all superintuitionistic logics, the uniform interpolation property is equivalent to Craig's property.On elimination of quantifiers in some non-classical mathematical theorieshttps://zbmath.org/1521.030412023-11-13T18:48:18.785376Z"Badia, Guillermo"https://zbmath.org/authors/?q=ai:badia.guillermo"Tedder, Andrew"https://zbmath.org/authors/?q=ai:tedder.andrewSummary: Elimination of quantifiers is shown to fail dramatically for a group of well-known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.Derivation of preservation conditions for properties of mathematical modelshttps://zbmath.org/1521.030602023-11-13T18:48:18.785376Z"Vasil'ev, S. N."https://zbmath.org/authors/?q=ai:vasilev.stanislav-nikolaevich.1"Druzhinin, A. È."https://zbmath.org/authors/?q=ai:druzhinin.andrei-e"Morozov, N. Yu."https://zbmath.org/authors/?q=ai:morozov.n-yuSummary: A general method for deriving conditions for the preservation of properties of any mathematical models is proposed. No constraints on the properties under consideration are imposed. No a priori specification of any part of these conditions, except connection predicates for pairs of variables corresponding to each other in the definitions of properties, is required. Examples are given.Ehrenfeucht-Fraïssé games without identityhttps://zbmath.org/1521.030612023-11-13T18:48:18.785376Z"Urquhart, Alasdair"https://zbmath.org/authors/?q=ai:urquhart.alasdairSummary: Ehrenfeucht-Fraïssé games are usually formulated for a language including identity. In this note, we develop a formulation of the games for languages without identity. The new version is used to show that the identity relation on a structure cannot be characterized if identity is missing in the language.Automorphism invariant measures and weakly generic automorphismshttps://zbmath.org/1521.030622023-11-13T18:48:18.785376Z"Sági, Gábor"https://zbmath.org/authors/?q=ai:sagi.gaborSummary: Let \(\mathcal{A}\) be a countable \(\aleph_0\)-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group \(\Aut(\mathcal{A})\) of \(\mathcal{A}\); the secondary motivation is to study the existence of weakly generic automorphisms of \(\mathcal{A}\). Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of \(A\) and of \(\Aut(\mathcal{A})\); we also present sufficient conditions implying that the theory of \(\mathcal{A}\) is amenable. More concretely, we show that if the set of locally finite automorphisms of \(\mathcal{A}\) is dense (in particular, if \(\mathcal{A}\) has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure \(\mu\) on the subsets of \(\mathcal{A}\) definable with parameters such that \(\mu\) is invariant under \(\Aut(\mathcal{A})\). Moreover, if \(\mathcal{A}\) is saturated and the set of its locally finite automorphisms is dense (in particular, if \(\mathcal{A}\) is saturated and has weak generics), then the theory of \(\mathcal{A}\) is amenable.
{{\copyright} 2022 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH.}Interpreting the weak monadic second order theory of the ordered rationalshttps://zbmath.org/1521.030632023-11-13T18:48:18.785376Z"Truss, John K."https://zbmath.org/authors/?q=ai:truss.john-kennethSummary: We show that the weak monadic second order theory of the structure \(({\mathbb{Q}}, <)\) is first order interpretable in its automorphism group.
{{\copyright} 2021 Wiley-VCH GmbH}Atomic saturation of reduced powershttps://zbmath.org/1521.030642023-11-13T18:48:18.785376Z"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize ``the theory of dense linear order (or \(T\) with the strict order property) is maximal and so is any pair \(( T , \Delta)\) which is \(\mathrm{SOP}_3\)'', (where \(\Delta\) consists of atomic or conjunction of atomic formulas). However, the theorem on ``it is enough to deal with symmetric pre-cuts'' (so the \(\mathfrak{p} = \mathfrak{t}\) theorem) cannot be generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context.
{{\copyright} 2021 Wiley-VCH GmbH}Categoricity and universal classeshttps://zbmath.org/1521.030652023-11-13T18:48:18.785376Z"Hyttinen, Tapani"https://zbmath.org/authors/?q=ai:hyttinen.tapani"Kangas, Kaisa"https://zbmath.org/authors/?q=ai:kangas.kaisaSummary: Let \(\mathcal K, \subseteq\) be a universal class with \(\mathrm{LS}(\mathcal{K}) = \lambda\) categorical in a regular \(\kappa > \lambda^{+}\) with arbitrarily large models, and let \(\mathcal{K}{*}\) be the class of all \(\mathfrak{U} \in \mathcal K_{> \lambda}\) for which there is \(\mathfrak{B} \in \mathcal{K}_{\ge{\mathcal{K}}}\) such that \(\mathfrak{U} \subseteq \mathfrak{B}\). We prove that \(\mathcal{K}^{\ast}\) is totally categorical (i.e., \(\xi\)-categorical for all \(\xi > \mathrm{LS}(\mathcal{K})\)) and \(\mathcal K_{\ge \sqsupset_{\mu^+}} \subseteq \mathcal{K}^{\ast}\) for \(\mu=2^{\lambda^+}\). This result is partially stronger and partially weaker than a related result due to Vasey. In addition to small differences in our categoricity transfer results, we provide a shorter and simpler proof. In the end we prove the main theorem of this paper: the models of \(\mathcal{K}^{\ast}_{>\Lambda^+}\) are essentially vector spaces (or trivial, i.e., disintegrated).Generic expansion of an abelian variety by a subgrouphttps://zbmath.org/1521.030662023-11-13T18:48:18.785376Z"d'Elbée, Christian"https://zbmath.org/authors/?q=ai:delbee.christianSummary: Let \(A\) be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of \(A\) by a generic divisible subgroup of \(A\) with the same torsion exists provided \(A\) has few algebraic endomorphisms, namely \(\mathrm{End} (A) = \mathbb{Z}\). The resulting theory is \(\mathrm{NSOP}_1\) and not simple. Note that there exist abelian varieties \(A\) with \(\mathrm{End} (A) = \mathbb{Z}\) of any genus.
{{\copyright} 2021 Wiley-VCH GmbH}Non-forking and preservation of NIP and dp-rankhttps://zbmath.org/1521.030672023-11-13T18:48:18.785376Z"Estevan, Pedro Andrés"https://zbmath.org/authors/?q=ai:estevan.pedro-andres"Kaplan, Itay"https://zbmath.org/authors/?q=ai:kaplan.itaySummary: We investigate the question of whether the restriction of an NIP type \(p \in S(B)\) which does not fork over \(A\subseteq B\) to \(A\) is also NIP, and the analogous question for dp-rank. We show that if \(B\) contains a Morley sequence \(I\) generated by \(p\) over \(A\), then \(p\upharpoonright AI\) is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of ``trees whose open cones are models of some theory'' and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.Groups in \(\mathrm{NTP}_{2}\)https://zbmath.org/1521.030682023-11-13T18:48:18.785376Z"Hempel, Nadja"https://zbmath.org/authors/?q=ai:hempel.nadja"Onshuus, Alf"https://zbmath.org/authors/?q=ai:onshuus.alfSummary: We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an \(\mathrm{NTP}_{2}\) theory.Stability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaceshttps://zbmath.org/1521.030692023-11-13T18:48:18.785376Z"Khanaki, Karim"https://zbmath.org/authors/?q=ai:khanaki.karimSummary: We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, \textit{Talagrand's stability }, and explain the relationship between this property and the NIP in continuous logic. Using a result of \textit{J. Bourgain} et al. [Am. J. Math. 100, 845--886 (1978; Zbl 0413.54016)], we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula \(\varphi(x,y)\) has the strict order property if and only if there is a convergent sequence of continuous functions on the space of \(\phi\)-types such that its limit is not continuous. We deduce from this a theorem of \textit{S. Shelah} [Ann. Math. Logic 3, 271--362 (1971; Zbl 0281.02052)] and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.Remarks on the NIP in a modelhttps://zbmath.org/1521.030702023-11-13T18:48:18.785376Z"Khanaki, Karim"https://zbmath.org/authors/?q=ai:khanaki.karim"Pillay, Anand"https://zbmath.org/authors/?q=ai:pillay.anandSummary: We define the notion \(\varphi(x,y)\) has the NIP (not the independence property) in \(A\), where \(A\) is a subset of a model, and give some equivalences by translating results from function theory. We also discuss the number of coheirs when \(A\) is not necessarily countable, and revisit the notion ``\(\varphi(x,y)\) has the NOP (not the order property) in a model \(M\)''.Strongly NIP almost real closed fieldshttps://zbmath.org/1521.030712023-11-13T18:48:18.785376Z"Krapp, Lothar Sebastian"https://zbmath.org/authors/?q=ai:krapp.lothar-sebastian"Kuhlmann, Salma"https://zbmath.org/authors/?q=ai:kuhlmann.salma"Lehéricy, Gabriel"https://zbmath.org/authors/?q=ai:lehericy.gabrielSummary: The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.
{{\copyright} 2021 Wiley-VCH GmbH}Weakly binary expansions of dense meet-treeshttps://zbmath.org/1521.030722023-11-13T18:48:18.785376Z"Mennuni, Rosario"https://zbmath.org/authors/?q=ai:mennuni.rosarioSummary: We compute the domination monoid in the theory \(\mathsf{DMT}\) of dense meet-trees. In order to show that this monoid is well-defined, we prove \textit{weak binarity} of \(\mathsf{DMT}\) and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory \(\mathsf{DTR}\) from [\textit{P. A. Estevan} and \textit{I. Kaplan}, Ann. Pure Appl. Logic 172, No. 6, Article ID 102946, 30 p. (2021; Zbl 1521.03067)]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.
{{\copyright} 2021 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH}A note on the non-forking-instances topologyhttps://zbmath.org/1521.030732023-11-13T18:48:18.785376Z"Shami, Ziv"https://zbmath.org/authors/?q=ai:shami.zivSummary: The \textit{non-forking-instances topology} (NFI topology) is a topology on the Stone space of a theory \(T\) that depends on a reduct \(T^-\) of \(T\). This topology has been used in [\textit{Z. Shami}, Arch. Math. Logic 59, No. 3--4, 313--324 (2020; Zbl 1481.03015)] to describe the set of universal transducers for \(( T , T^- )\) (invariants sets that translate forking-open sets in \(T^-\) to forking-open sets in \(T)\). In this paper we show that in contrast to the stable case, the NFI topology need not be invariant over parameters in \(T^-\) but a weak version of this holds for any simple \(T\). We also note that for the lovely pair expansions, of theories with the weak non-finite cover property (wnfcp), the topology is invariant over \(\emptyset\) in \(T^-\).
{{\copyright} 2020 Wiley-VCH GmbH}Some model theory of \(\mathrm{Th}(\mathbb{N},\cdot)\)https://zbmath.org/1521.030742023-11-13T18:48:18.785376Z"Stonestrom, Atticus"https://zbmath.org/authors/?q=ai:stonestrom.atticusSummary: `Skolem arithmetic' is the complete theory \(T\) of the multiplicative monoid \((\mathbb{N},\cdot)\). We give a full characterization of the \(\emptyset\)-definable stably embedded sets of \(T\), showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that \(T\) has weak elimination of imaginaries but not elimination of finite imaginaries.
{{\copyright} 2022 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH.}On definability of types and relative stabilityhttps://zbmath.org/1521.030752023-11-13T18:48:18.785376Z"Verbovskiy, Viktor"https://zbmath.org/authors/?q=ai:verbovskii.viktor-valerievichSummary: In this paper, we consider the question of definability of types in non-stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to \(\Delta\) if any \(\Delta\)-type over a model has few extensions up to complete types. We prove that an \(n\)-type over a model of a theory that is stable up to \(\Delta\) is definable if and only if its \(\Delta\)-part is definable.Shelah's eventual categoricity conjecture in tame abstract elementary classes with primeshttps://zbmath.org/1521.030762023-11-13T18:48:18.785376Z"Vasey, Sebastien"https://zbmath.org/authors/?q=ai:vasey.sebastienSummary: A new case of Shelah's eventual categoricity conjecture is established:
Let \(\mathcal{K}\) be an abstract elementary class with amalgamation. Write \(\mu:= \beth_{(2^{\mathrm{LS}(\mathcal{K})})^{+}}\) and \(H_2:= \beth_{(2^{\mu})^{+}}\). Assume that \(\mathcal{K}\) is \(H_2\)-tame and \(\mathcal{K}_{\geq H_{2}}\) has primes over sets of the form \(M\cup\{a\}\). If \(\mathcal{K}\) is categorical in some \(\lambda>H_2\), then \(\mathcal{K}\) is categorical in all \(\lambda'\geq H_2\).
The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If \(D\) be a homogeneous diagram in a first-order theory \(T\) and \(D\) is categorical in a \(\lambda>|T|\), then \(D\) is categorical in all \(\lambda'\geq \min (\lambda, \beth_{(2^{|T|})^{+}})\).The theory of hereditarily bounded setshttps://zbmath.org/1521.030772023-11-13T18:48:18.785376Z"Jeřábek, Emil"https://zbmath.org/authors/?q=ai:jerabek.emilSummary: We show that for any \(k\in \omega \), the structure \(\langle H_k,{\in }\rangle\) of sets that are hereditarily of size at most \(k\) is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure \(V_\omega =\bigcup_kH_k\) of hereditarily finite sets, which is well known to be bi-interpretable with the standard model of arithmetic \(\langle \mathbb{N},+,\cdot \rangle \).
{{\copyright} 2022 Wiley-VCH GmbH}Examples of weak amalgamation classeshttps://zbmath.org/1521.030782023-11-13T18:48:18.785376Z"Krawczyk, Adam"https://zbmath.org/authors/?q=ai:krawczyk.adam"Kruckman, Alex"https://zbmath.org/authors/?q=ai:kruckman.alex"Kubiś, Wiesław"https://zbmath.org/authors/?q=ai:kubis.wieslaw"Panagiotopoulos, Aristotelis"https://zbmath.org/authors/?q=ai:panagiotopoulos.aristotelisSummary: We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum-sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.
{{\copyright} 2022 Wiley-VCH GmbH}The profinite topology of free groups and weakly generic tuples of automorphismshttps://zbmath.org/1521.030792023-11-13T18:48:18.785376Z"Sági, Gábor"https://zbmath.org/authors/?q=ai:sagi.gaborSummary: Let \(\mathcal{A}\) be a countable first order structure and endow the universe of \(\mathcal{A}\) with the discrete topology. Then the automorphism group \(\Aut (\mathcal{A})\) of \(\mathcal{A}\) becomes a topological group. A tuple of automorphisms \(\langle g_0, \ldots, g_{n - 1} \rangle\) is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the \(\langle g_0, \ldots, g_{n - 1} \rangle\)-orbit of each \(a \in A\) is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics. The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if \(\Aut (\mathcal{A})\) has finite topological rank \(r\) (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in \(\Aut (\mathcal{A})^r\) implies the existence of weakly generic tuples in \(\Aut (\mathcal{A})^n\) for all natural number \(n \geq 1\). Finally, in Theorem 3.2 we show that if \(\mathcal{A}\) is a countable model of an \(\aleph_0\)-categorical, simple theory in which all types over the empty set are stationary and \(\mathcal{A}\) has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length. At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [\textit{I. Kaplan} and \textit{P. Simon}, Trans. Am. Math. Soc. 372, No. 3, 2011--2043 (2019; Zbl 1506.20001)] about topological ranks of the automorphism groups of some structures.
{{\copyright} 2021 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH}Ultrafilter extensions do not preserve elementary equivalencehttps://zbmath.org/1521.030802023-11-13T18:48:18.785376Z"Saveliev, Denis I."https://zbmath.org/authors/?q=ai:saveliev.denis-i"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: We show that there are models \(\mathcal{M}_1\) and \(\mathcal{M}_2\) such that \(\mathcal{M}_1\) elementarily embeds into \(\mathcal{M}_2\) but their ultrafilter extensions \(\beta\beta(\mathcal{M}_1)\) and \(\beta\beta(\mathcal{M}_2)\) are not elementarily equivalent.The gap-two cardinal problem for uncountable languageshttps://zbmath.org/1521.030812023-11-13T18:48:18.785376Z"Villegas Silva, Luis Miguel"https://zbmath.org/authors/?q=ai:villegas-silva.luis-miguelSummary: In this paper we prove some cases of the gap-2 cardinal transfer theorem for uncountable languages assuming the axiom of constructibility. Consider uncountable cardinals \(\kappa\), \(\lambda\) with \(\lambda\) regular, a first order language \(\mathcal{L}\) with at least one unary predicate symbol \(U\), with \(\vert \mathcal{L}\vert < \min\{\kappa, \lambda\}\). Given an \(\mathcal{L}\)-structure \(\mathfrak{U} = \langle A,U^{\mathfrak{U}}, \ldots \rangle\), where \(\vert A \vert = \kappa^{++}\) and \(\vert U ^{\mathfrak{U}}\vert = \kappa\), we find an \(\mathcal{L}\)-structure \(\mathfrak{B} = (B, U^{\mathfrak{U}}, \ldots)\) such that \(\mathfrak{B} \equiv \mathfrak{U}\), \(\vert B\vert = \lambda^{++}\) and \(U^{\mathfrak{U}} = \lambda\).On the effective universality of mereological theorieshttps://zbmath.org/1521.030822023-11-13T18:48:18.785376Z"Bazhenov, Nikolay"https://zbmath.org/authors/?q=ai:bazhenov.n-a"Tsai, Hsing-Chien"https://zbmath.org/authors/?q=ai:tsai.hsing-chienSummary: Mereological theories are based on the binary relation ``being a part of''. The systematic investigations of mereology were initiated by \textit{S. Leśniewski} [Collected works. Volume I and II. Edited by Stanisław J. Surma, Jan T. Srzednicki and D. I. Barnett. With an annotated bibliography by V. Frederick Rickey. Dordrecht: Kluwer Academic Publishers; Warsaw: PWN-Polish Scientific Publishers (1992; Zbl 0765.03002)]. More recent authors (including \textit{P. Simons} [Parts: a Study in ontology. Oxford: Clarendon Press (1987)], \textit{A. Casati} and \textit{A. C. Varzi} [Parts and places. Cambridge: The MIT Press (1999)], \textit{P. Hovda} [J. Philos. Log. 38, No. 1, 55--82 (2009; Zbl 1171.03002)]) formulated a series of first-order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first-order mereological theories from the point of view of computable (or effective) algebra. Following the approach of \textit{D. R. Hirschfeldt} et al. [Ann. Pure Appl. Logic 115, No. 1--3, 71--113 (2002; Zbl 1016.03034)], we isolate two important computability-theoretic properties \(\mathrm{P}\) (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory \(T\), is it true that its models can realize every possible variant of the property \(\mathrm{P}\)? If the answer is positive, then we say that the theory \(T\) is \(\mathit{DSED} \)-universal. We obtain the following results about known mereological theories. Any theory \(T\) which is weaker than Extensional Closure Mereology (CEM) is \(\mathit{DSED} \)-universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not \(\mathit{DSED} \)-universal. In particular, General Extensional Mereology is not \(\mathit{DSED} \)-universal.
{{\copyright} 2021 Wiley-VCH GmbH}Jump inversions of algebraic structures and \(\Sigma \)-definabilityhttps://zbmath.org/1521.030832023-11-13T18:48:18.785376Z"Faizrahmanov, Marat"https://zbmath.org/authors/?q=ai:faizrahmanov.marat-khaidarovich"Kach, Asher"https://zbmath.org/authors/?q=ai:kach.asher-m"Kalimullin, Iskander"https://zbmath.org/authors/?q=ai:kalimullin.iskander-sh"Montalbán, Antonio"https://zbmath.org/authors/?q=ai:montalban.antonio"Puzarenko, Vadim"https://zbmath.org/authors/?q=ai:puzarenko.vadim-gSummary: It is proved that for every countable structure \(\mathcal{A}\) and a computable successor ordinal \(\alpha\) there is a countable structure \(\mathcal{A}^{-\alpha}\) which is \(\leq _{\Sigma}\)-least among all countable structures \(\mathcal{C}\) such that \(\mathcal{A}\) is \(\Sigma\)-definable in the \(\alpha\)th jump \(\mathcal{C}^{(\alpha)}\). We also show that this result does not hold for the limit ordinal \(\alpha=\omega\). Moreover, we prove that there is no countable structure \(\mathcal{A}\) with the degree spectrum \(\{\mathbf{d}:\mathbf{a}\leq\mathbf{d}^{(\omega)}\}\) for \(\mathbf{a}>\mathbf{0}^{(\omega)}\).Bi-embeddability spectra and bases of spectrahttps://zbmath.org/1521.030842023-11-13T18:48:18.785376Z"Fokina, Ekaterina"https://zbmath.org/authors/?q=ai:fokina.ekaterina-b"Rossegger, Dino"https://zbmath.org/authors/?q=ai:rossegger.dino"San Mauro, Luca"https://zbmath.org/authors/?q=ai:san-mauro.lucaSummary: We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.Degrees of categoricity of trees and the isomorphism problemhttps://zbmath.org/1521.030852023-11-13T18:48:18.785376Z"Mahmoud, Mohammad Assem"https://zbmath.org/authors/?q=ai:mahmoud.mohammad-assemSummary: In this paper, we show that for any computable ordinal \(\alpha\), there exists a computable tree of rank \(\alpha+1\) with strong degree of categoricity \(0^{(2\alpha)}\) if \(\alpha\) is finite, and with strong degree of categoricity \(0^{(2\alpha+1)}\) if \(\alpha\) is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal \(\alpha\), we show that there is a computable tree of rank \(\alpha\) with strong degree of categoricity \(0^{(\alpha)}\) (which equals \(0^{(2\alpha)}\)). It follows from our proofs that, for every computable ordinal \(\alpha>0\), the isomorphism problem for trees of rank \(\alpha\) is \(\Pi_{2\alpha}^0\)-complete.On differential Galois groups of strongly normal extensionshttps://zbmath.org/1521.030862023-11-13T18:48:18.785376Z"Brouette, Quentin"https://zbmath.org/authors/?q=ai:brouette.quentin"Point, Françoise"https://zbmath.org/authors/?q=ai:point.francoiseSummary: We revisit \textit{E. R. Kolchin}'s [Differential algebra and algebraic groups. New York, NY: Academic Press (1973; Zbl 0264.12102)] results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or \(p\)-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in \textsf{CODF}, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorphisms.The torsion-free part of the Ziegler spectrum of orders over Dedekind domainshttps://zbmath.org/1521.030872023-11-13T18:48:18.785376Z"Gregory, Lorna"https://zbmath.org/authors/?q=ai:gregory.lorna"L'Innocente, Sonia"https://zbmath.org/authors/?q=ai:linnocente.sonia"Toffalori, Carlo"https://zbmath.org/authors/?q=ai:toffalori.carloSummary: We study the \(R\)-torsion-free part of the Ziegler spectrum of an order \(\Lambda\) over a Dedekind domain \(R\). We underline and comment on the role of lattices over \(\Lambda \). We describe the torsion-free part of the spectrum when \(\Lambda\) is of finite lattice representation type.On universal modules with pure embeddingshttps://zbmath.org/1521.030882023-11-13T18:48:18.785376Z"Kucera, Thomas G."https://zbmath.org/authors/?q=ai:kucera.thomas-g"Mazari-Armida, Marcos"https://zbmath.org/authors/?q=ai:mazari-armida.marcosSummary: We show that certain classes of modules have universal models with respect to pure embeddings: Let \(R\) be a ring, \(T\) a first-order theory with an infinite model extending the theory of \(R\)-modules and \(\mathbf{K}^T = (\mathrm{Mod} (T), \leq_{\mathrm{pp}})\) (where \(\leq_{pp}\) stands for ``pure submodule''). Assume \(\mathbf{K}^T\) has the joint embedding and amalgamation properties. If \(\lambda^{| T |} = \lambda\) or \(\forall \mu < \lambda (\mu^{| T |} < \lambda)\), then \(\mathbf{K}^T\) has a universal model of cardinality \(\lambda\). As a special case, we get a recent result of \textit{S. Shelah} [Notre Dame J. Formal Logic 58, No. 2, 159--177 (2017; Zbl 1417.03231), 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of \(R\)-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer [\textit{M. Mazari-Armida}, Ann. Pure Appl. Logic 171, No. 1, Article ID 102723, 17 p. (2020; Zbl 1480.03019), Question 4.25]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.
{{\copyright} 2021 Wiley-VCH GmbH}Pseudo-C-Archimedean and pseudo-finite cyclically ordered groupshttps://zbmath.org/1521.030892023-11-13T18:48:18.785376Z"Leloup, Gérard"https://zbmath.org/authors/?q=ai:leloup.gerardSummary: \textit{A. Robinson} and \textit{E. Zakon} [Trans. Am. Math. Soc. 96, 222--236 (1960; Zbl 0096.24504)] gave necessary and sufficient conditions for an abelian ordered group to satisfy the same first-order sentences as an archimedean abelian ordered group (i.e., which embeds in the group of real numbers). The present paper generalizes their work to obtain similar results for infinite subgroups of the group of unimodular complex numbers. Furthermore, the groups which satisfy the same first-order sentences as ultraproducts of finite cyclic groups are characterized.Expansions of the \(p\)-adic numbers that interpret the ring of integershttps://zbmath.org/1521.030902023-11-13T18:48:18.785376Z"Mariaule, Nathanaël"https://zbmath.org/authors/?q=ai:mariaule.nathanaelSummary: Let \(\widetilde{\mathbb{Q}_p}\) be the field of \(p\)-adic numbers in the language of rings. In this paper we consider the theory of \(\widetilde{\mathbb{Q}_p}\) expanded by two predicates interpreted by multiplicative subgroups \(\alpha^{\mathbb{Z}}\) and \(\beta^{\mathbb{Z}}\) where \(\alpha, \beta \in \mathbb{N}\) are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if \(\alpha\) and \(\beta\) have positive \(p\)-adic valuation. If either \(\alpha\) or \(\beta\) has zero valuation we show that the theory of \((\widetilde{\mathbb{Q}_p}, \alpha^{\mathbb{Z}}, \beta^{\mathbb{Z}})\) has the NIP (``negation of the independence property'') and therefore does not interpret Peano arithmetic. In that case we also prove that the theory is decidable if and only if the theory of \((\widetilde{\mathbb{Q}_p}, \alpha^{\mathbb{Z}} \beta^{\mathbb{Z}})\) is decidable.Éz fieldshttps://zbmath.org/1521.030912023-11-13T18:48:18.785376Z"Walsberg, Erik"https://zbmath.org/authors/?q=ai:walsberg.erik"Ye, Jinhe"https://zbmath.org/authors/?q=ai:ye.jinheThe paper is concerned with two properties of fields, largeness and logical tameness. Largeness is a precise notion: a field \(K\) is large if every curve with a smooth \(K\)-point contains inifnitely many \(K\)-points. Logical tameness is not a precise notion; the authors interpret it roughly as some sort of quantifier elimination.
In a previous paper \textit{W. Johnson} et al. [``The étale-open topology and the stable fields conjecture'', J. Eur. Math. Soc. (JEMS) (to appear), \url{doi:10.4171/JEMS/1345}] introduced the étale open topology on fields. The étale open topology on \(V(K)\), the set of \(K\)-points of a variety \(V\), has a basis of open sets of the form \(f(W(K))\) for étale morphisms \(f:W \to K\). Then \(K\) is large if and only if this topology on the affine line over \(K\) is not discrete. If \(K\) is separably, real, \(p\)-adically closed then the étale open topology agrees with the Zariski, order, valuation topology, respectively. The authors show that if \(K\) is perfect and large, then existentially definable sets behave well with respect to the étale open topology. In particular, if \(K\) is model complete in the language of rings then all definable sets are well-behaved with respect to this topology.
In order to state the main results of the paper, we need to introduce the notion of éz subsets first. An éz subset of \(V(K)\) is a finite union of definable étale open subsets of Zariski closed subsets of \(V(K)\). A field \(K\) is said to be éz if it is large and every definable set is an éz set. This can be viewed as a topological generalization of model completeness for large fields. Algebraically closed, real closed, \(p\)-adically closed, and bounded PAC fields are éz.
The first main result of the paper is as follows.
Theorem A. Suppose \(K\) is large and perfect and \(f : V \to W\) is a morphism of \(K\)-varieties. If \(X\) is an éz subset of \(V(K)\) then \(f(X)\) is an éz subset of \(W(K)\).
This can be seen as an analogue of Chevalley's theorem stating that over an algebraically closed field the image of a constructible set under a morphism is constructible (which is equivalent to quantifier elimination). Theorem A actually generalises this statement, for the étale open topology over an algebraically closed field agrees with the Zariski topology.
Corollary A. Let \(K\) be large and perfect. Then any existentially definable subset of any \(K^m\) is an éz set. In particular, any existentially definable subset of \(K\) is a union of a definable étale open subset of \(K\) and a finite set.
Theorem B. is a structural theorem for éz subsets. It states in particular that over a large perfect field \(K\), given a smooth variety \(V\) and its decomposititon into a disjoint union of irreducible components, any non-empty éz subset of \(V(K)\) can be decomposed into a union of definable étale open subsets of the components of \(V\). Further, \(\dim X = \dim V\) if and only if in the étale open topology \(X\) has non-empty interior as a subset of \(V(K)\), and this remains true if we replace \(V(K)\) with an éz subset \(Y\) containing \(X\).
Theorem C. Fields with one of the following peoperties are éz.
\begin{itemize}
\item \(K\) is large and model complete.
\item \(K\) is large, perfect, and model complete after naming some constants.
\item \(K\) is \(t\)-Henselian of characteristic zero.
\item \(K\) is a perfect Frobenius field.
\end{itemize}
Theorem D. Éz fields are algebraically bounded.
Since algebraically bounded fields are geometric (i.e. they eliminate \(\exists^{\infty}\), the model-theoretic algebraic closure has the exchange property, and the resulting notion of dimension agrees with algebraic dimension), the authors are able to deduce a fibre dimension theorem for definable sets in éz fields from Theorem D.
Theorem E. Let \(K\) be an éz field and let \(f : K^m \to K^n\) be definable. Then \(f\) is continuous with respect to the étale open topology on a dense Zariski open subset of \(K^m\).
The authors also discuss some conjecture and open questions. In particular, combining a question of Macintyre and a conjecture of Koenigsmann, they ask whether every model complete field is large or, equivalently, whether every model complete field is éz.
Reviewer: Vahagn Aslanyan (Manchester)Definable topological dynamics for trigonalizable algebraic groups over \(\mathbb{Q}_P\)https://zbmath.org/1521.030922023-11-13T18:48:18.785376Z"Yao, Ningyuan"https://zbmath.org/authors/?q=ai:yao.ningyuanSummary: We study the flow \((G(\mathbb{Q}_p),S_G(\mathbb{Q}_p))\) of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [\textit{A. Pillay} and the author, Adv. Math. 290, 483--502 (2016; Zbl 1386.03042)] of whether weakly generic types coincide with almost periodic types if the group has global definable f-generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f-generic types of trigonalizable algebraic groups, and prove that every f-generic type is almost periodic.Neutrally expandable models of arithmetichttps://zbmath.org/1521.030932023-11-13T18:48:18.785376Z"Abdul-Quader, Athar"https://zbmath.org/authors/?q=ai:abdul-quader.athar"Kossak, Roman"https://zbmath.org/authors/?q=ai:kossak.romanSummary: A subset of a model of PA is called \textit{neutral} if it does not change the dcl relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non-existence of neutral sets in various models of PA. We show that cofinal extensions of prime models are neutrally expandable, and \(\omega_1\)-like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first-order property. In the last section, we study a local version of neutral expandability.CP-generic expansions of models of Peano arithmetichttps://zbmath.org/1521.030942023-11-13T18:48:18.785376Z"Abdul-Quader, Athar"https://zbmath.org/authors/?q=ai:abdul-quader.athar"Schmerl, James H."https://zbmath.org/authors/?q=ai:schmerl.james-hSummary: We study notions of genericity in models of \(\mathsf{PA} \), inspired by lines of inquiry initiated by \textit{Z. Chatzidakis} and \textit{A. Pillay} [Ann. Pure Appl. Logic 95, No. 1--3, 71--92 (1998; Zbl 0929.03043)] and continued by \textit{A. Dolich} et al. [Ann. Pure Appl. Logic 167, No. 8, 684--706 (2016; Zbl 1432.03070)] in general model-theoretic contexts. These papers studied the theories obtained by adding a ``random'' predicate to a class of structures. Chatzidakis and Pillay [loc. cit.] axiomatized the theories obtained in this way. In this article, we look at the subsets of models of \(\mathsf{PA}\) which satisfy the axiomatization given by Chatzidakis and Pillay [loc. cit.]; we refer to these subsets in models of \(\mathsf{PA}\) as CP-generics. We study a more natural property, called strong CP-genericity, which implies CP-genericity. We use an arithmetic version of Cohen forcing to construct (strong) CP-generics with various properties, including ones in which every element of the model is definable in the expansion, and, on the other extreme, ones in which the definable closure relation is unchanged.
{{\copyright} 2022 Wiley-VCH GmbH}Episodes in model-theoretic xenology: rationals as positive integers in \textsf{R}\(^\sharp\)https://zbmath.org/1521.030952023-11-13T18:48:18.785376Z"Ferguson, Thomas Macaulay"https://zbmath.org/authors/?q=ai:ferguson.thomas-macaulay"Ramírez-Cámara, Elisángela"https://zbmath.org/authors/?q=ai:ramirez-camara.elisangelaSummary: \textit{R. K. Meyer} and \textit{C. Mortensen}'s [ibid. 18, No. 5, 401--425 (2021; Zbl 1521.03229)] \textit{Alien intruder theorem} includes the extraordinary observation that the rationals can be extended to a model of the relevant arithmetic \textsf{R}\(^\sharp\), thereby serving as integers themselves. Although the mysteriousness of this observation is acknowledged, little is done to explain \textit{why} such rationals-as-integers exist or \textit{how} they operate. In this paper, we show that Meyer and Mortensen's models can be identified with a class of ultraproducts of finite models of \textsf{R}\(^\sharp\), providing insights into some of the more mysterious phenomena of the rational models.Expansions of Presburger arithmetic with the exchange propertyhttps://zbmath.org/1521.030962023-11-13T18:48:18.785376Z"Mariaule, Nathanaël"https://zbmath.org/authors/?q=ai:mariaule.nathanaelSummary: Let \(G\) be a model of Presburger arithmetic. Let \(\mathcal{L}\) be an expansion of the language of Presburger \(\mathcal{L}_{\mathrm{Pres}}\). In this paper, we prove that the \(\mathcal{L}\)-theory of \(G\) is \(\mathcal{L}_{\mathrm{Pres}}\)-minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the \(\mathcal{L}\)-theory of \(G\) has the exchange property but is not definably complete, there is a proper definable convex subgroup \(H\). Assuming that the induced theories on \(H\) and \(G / H\) are definable complete and \(o\)-minimal respectively, we prove that any definable set of \(G\) is \(\mathcal{L}_{\mathrm{Pres}} \cup \{ H \}\)-definable.
{{\copyright} 2021 Wiley-VCH GmbH}Pathological examples of structures with o-minimal open corehttps://zbmath.org/1521.030972023-11-13T18:48:18.785376Z"Block Gorman, Alexi"https://zbmath.org/authors/?q=ai:gorman.alexi-block"Caulfield, Erin"https://zbmath.org/authors/?q=ai:caulfield.erin"Hieronymi, Philipp"https://zbmath.org/authors/?q=ai:hieronymi.philippSummary: This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure \(\mathcal{R}\) by a unary predicate such that its open core is a proper o-minimal expansion of \(\mathcal{R}\). We give an example of a structure that has an o-minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o-minimal open core and definable Skolem functions, but is not o-minimal.
{{\copyright} 2021 Wiley-VCH GmbH}Product cones in dense pairshttps://zbmath.org/1521.030982023-11-13T18:48:18.785376Z"Eleftheriou, Pantelis E."https://zbmath.org/authors/?q=ai:eleftheriou.pantelis-eSummary: Let \(\mathcal{M}=\langle M, <, +, \dots \rangle\) be an o-minimal expansion of an ordered group, and \(P\subseteq M\) a dense set such that certain tameness conditions hold. We introduce the notion of a \textit{product cone} in \(\widetilde{\mathcal{M}}=\langle \mathcal{M}, P\rangle\), and prove: if \(\mathcal{M}\) expands a real closed field, then \(\widetilde{\mathcal{M}}\) admits a product cone decomposition. If \(\mathcal{M}\) is linear, then it does not. In particular, we settle a question from [10].
{{\copyright} 2022 The Authors. Mathematical Logic Quarterly published by Wiley-VCH GmbH.}The choice property in tame expansions of o-minimal structureshttps://zbmath.org/1521.030992023-11-13T18:48:18.785376Z"Eleftheriou, Pantelis E."https://zbmath.org/authors/?q=ai:eleftheriou.pantelis-e"Günaydın, Ayhan"https://zbmath.org/authors/?q=ai:gunaydin.ayhan"Hieronymi, Philipp"https://zbmath.org/authors/?q=ai:hieronymi.philippSummary: We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, this property holds for dense pairs of real closed fields, as well as for expansions of o-minimal structures by a dense independent set.Uniformly locally o-minimal open corehttps://zbmath.org/1521.031002023-11-13T18:48:18.785376Z"Fujita, Masato"https://zbmath.org/authors/?q=ai:fujita.masato|fujita.masato.1Summary: This paper discusses sufficient conditions for a definably complete expansion of a densely linearly ordered abelian group to have uniformly locally o-minimal open cores of the first/second kind and strongly locally o-minimal open core, respectively.
{{\copyright} 2021 Wiley-VCH GmbH}Tameness of definably complete locally o-minimal structures and definable bounded multiplicationhttps://zbmath.org/1521.031012023-11-13T18:48:18.785376Z"Fujita, Masato"https://zbmath.org/authors/?q=ai:fujita.masato|fujita.masato.1"Kawakami, Tomohiro"https://zbmath.org/authors/?q=ai:kawakami.tomohiro"Komine, Wataru"https://zbmath.org/authors/?q=ai:komine.wataruSummary: We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies a tame dimension theory and a decomposition theorem into good-shaped definable subsets called quasi-special submanifolds. Using this fact, we investigate definably complete locally o-minimal expansions of ordered groups when the restriction of multiplication to an arbitrary bounded open box is definable. Similarly to o-minimal expansions of ordered fields, Łojasiewicz's inequality, Tietze's extension theorem and affiness of pseudo-definable spaces hold true for such structures under the extra assumption that the domains of definition and the pseudo-definable spaces are definably compact. Here, a pseudo-definable space is a topological space having finite definable atlases. We also demonstrate Michael's selection theorem for definable set-valued functions with definably compact domains of definition.
{{\copyright} 2022 Wiley-VCH GmbH.}Distal and non-distal behavior in pairshttps://zbmath.org/1521.031022023-11-13T18:48:18.785376Z"Nell, Travis"https://zbmath.org/authors/?q=ai:nell.travisSummary: The aim of this work is an analysis of distal and non-distal behavior in dense pairs of o-minimal structures. A characterization of distal types is given through orthogonality to a generic type in \(\mathcal{M}^{\mathrm{cq}}\), non-distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.On \(p\)-adic semi-algebraic continuous selectionshttps://zbmath.org/1521.031032023-11-13T18:48:18.785376Z"Thamrongthanyalak, Athipat"https://zbmath.org/authors/?q=ai:thamrongthanyalak.athipatSummary: Let \(E \subseteq \mathbb{Q}_p^n\) and \(T\) be a set-valued map from \(E\) to \(\mathbb{Q}_p^m\). We prove that if \(T\) is \(p\)-adic semi-algebraic, lower semi-continuous and \(T (x)\) is closed for every \(x \in E\), then \(T\) has a \(p\)-adic semi-algebraic continuous selection. In addition, we include three applications of this result. The first one is related to \textit{C. Fefferman}'s and \textit{J. Kollár}'s [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)] question on existence of \(p\)-adic semi-algebraic continuous solution of linear equations with polynomial coefficients. The second one is about the existence of \(p\)-adic semi-algebraic continuous extensions of continuous functions. The other application is on the characterization of right invertible \(p\)-adic semi-algebraic continuous functions under the composition.Concrete barriers to quantifier elimination in finite dimensional \(C^*\)-algebrashttps://zbmath.org/1521.031042023-11-13T18:48:18.785376Z"Eagle, Christopher J."https://zbmath.org/authors/?q=ai:eagle.christopher-james|eagle.christopher-j"Schmid, Todd"https://zbmath.org/authors/?q=ai:schmid.toddSummary: Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati [the first author et al., Int. Math. Res. Not. 2017, No. 24, 7580--7606 (2017; Zbl 1404.03029); Topology Appl. 207, 1--9 (2016; Zbl 1338.54141)] shows that the only separable \(C^*\)-algebras that admit quantifier elimination in continuous logic are \(\mathbb{C}\), \(\mathbb{C}^2\), \(M_2(\mathbb{C})\), and the continuous functions on the Cantor set. We show that, among finite dimensional \(C^*\)-algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate elements. Both of these predicates are definable, but not quantifier-free definable, in the usual language of \(C^*\)-algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.\(\aleph_0\)-categorical Banach spaces contain \(\ell_p\) or \(c_0\)https://zbmath.org/1521.031052023-11-13T18:48:18.785376Z"Khanaki, Karim"https://zbmath.org/authors/?q=ai:khanaki.karimSummary: This paper has three parts. First, we establish some of the basic model theoretic facts about \(M_{\mathcal{T}}\), the Tsirelson space of \textit{T. Figiel} and \textit{W. B. Johnson} [Compos. Math. 29, 179--190 (1974; Zbl 0301.46013)]. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In particular, we show: \((1) M_{\mathcal{T}}\) has the \textit{non independence property} (NIP); (2) every Banach space that is \(\aleph_0\)-categorical up to small perturbations embeds \(c_0\) or \(\ell_p (1 \leqslant p < \infty)\) almost isometrically; consequently the (continuous) first-order theory of \(M_{\mathcal{T}}\) does not characterize \(M_{\mathcal{T}}\), up to almost isometric isomorphism.
{{\copyright} 2021 Wiley-VCH GmbH}The isomorphism theorem for linear fragments of continuous logichttps://zbmath.org/1521.031062023-11-13T18:48:18.785376Z"Bagheri, Seyed-Mohammad"https://zbmath.org/authors/?q=ai:bagheri.seyed-mohammadSummary: The ultraproduct construction is generalized to \(p\)-ultramean constructions (\( 1 \leqslant p < \infty \)) by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments \(\mathcal{L}^p\) of continuous logic and are very close to the \(L^p ( \mathbb{R} )\) constructions in real analysis. A powermean variant of the Keisler-Shelah isomorphism theorem is proved for \(\mathcal{L}^p\). It is then proved that \(\mathcal{L}^p\)-sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.
{{\copyright} 2021 Wiley-VCH GmbH}Model-theoretic properties of dynamics on the Cantor sethttps://zbmath.org/1521.031072023-11-13T18:48:18.785376Z"Eagle, Christopher J."https://zbmath.org/authors/?q=ai:eagle.christopher-james|eagle.christopher-j"Getz, Alan"https://zbmath.org/authors/?q=ai:getz.alanThis paper uses the tools of continuous first-order logic to study dynamical systems of the form \(\langle X,s\rangle\), where \(X\) is a Cantor set and \(s\) is a homeomorphism on \(X\). By the Gel'fand duality, the category of compact Hausdorff spaces and continuous maps is contravariantly equivalent to the category of commutative unital \(C^*\)-algebras and unit-preserving algebra homomorphisms. This allows continuous model theory to be applied to structures of the form \(\langle C,\sigma\rangle\), where \(C=C(X)\), is the \(C^*\)-algebra of all continuous complex-valued functions on \(X\) (the Gel'fand dual of \(X\)), and \(\sigma=C(s)\) is the isomorphism on \(C\) induced by \(S\). The main results are the following:
\begin{itemize}
\item[Theorem 1.] The theory of \(\langle C,\sigma\rangle\) does not have a model companion.
\item[Theorem 2.] If two odometers on \(X\) are elementarily equivalent, then they are topologically conjugate.
\item[Theorem 3.] Being a generic homeomorphism is not axiomatizable, but it is expressible as an omitting types property. If \(s\) is the generic homeomorphism of the Cantor set \(X\), then \(\langle C,\sigma\rangle\) is the prime model of its theory.
\end{itemize}
Reviewer: Paul Bankston (Milwaukee)Unbounded actions of metric groups and continuous logichttps://zbmath.org/1521.031082023-11-13T18:48:18.785376Z"Ivanov, Aleksander"https://zbmath.org/authors/?q=ai:ivanov.aleksandr-vladimirovich|ivanov.aleksandr-pavlovich|ivanov.aleksandr-olegovich|ivanov.aleksandr-yurevich|ivanov.aleksandr-gennadevich|ivanov.aleksandr-aleksandrovich|ivanov.aleksandr-vladimirovich.1|ivanov.aleksander|ivanov.aleksandr-s|ivanov.aleksandr-valentinovich|ivanov.aleksandr-aleksandrovich.1Summary: We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. We concentrate on unbounded continuous actions on metric spaces. For example, we consider the properties non-\textbf{OB}, non-\textbf{FH} and non-\textbf{FR}.
{{\copyright} 2021 Wiley-VCH GmbH}Continuous theory of operator expansions of finite dimensional Hilbert spaces and decidabilityhttps://zbmath.org/1521.031092023-11-13T18:48:18.785376Z"Ivanov (Iwanow), Aleksander"https://zbmath.org/authors/?q=ai:ivanov.aleksander|ivanov.aleksandr-s|ivanov.aleksandr-valentinovich|ivanov.aleksandr-vladimirovich.1|ivanov.aleksandr-vladimirovich|ivanov.aleksandr-yurevich|ivanov.aleksandr-pavlovich|ivanov.aleksandr-gennadevich|ivanov.aleksandr-olegovich|ivanov.aleksandr-aleksandrovich.1|ivanov.aleksandr-aleksandrovichSummary: We consider continuous structures which are obtained from finite dimensional Hilbert spaces over \(\mathbf{C}\) by adding some unitary operators. We consider appropriate algorithmic problems concerning continuous theories of natural classes of these structures. We connect these questions with property MF.
{{\copyright} 2021 Wiley-VCH GmbH}Complete \(\mathcal{L}_{\omega_1,\omega}\)-sentences with maximal models in multiple cardinalitieshttps://zbmath.org/1521.031102023-11-13T18:48:18.785376Z"Baldwin, John"https://zbmath.org/authors/?q=ai:baldwin.john-t"Souldatos, Ioannis"https://zbmath.org/authors/?q=ai:souldatos.ioannis-aSummary: In [\textit{J. T. Baldwin} et al., Arch. Math. Logic 55, No. 3--4, 545--565 (2016; Zbl 1436.03190)], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete \(\mathcal{L}_{\omega_1,\omega}\)-sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of \(\kappa\) we construct sentences with maximal models in \(\kappa\) and in one of \(\kappa^+\), \(\kappa^\omega\), \(2^\kappa\) and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.The Hanf number in the strictly stable casehttps://zbmath.org/1521.031112023-11-13T18:48:18.785376Z"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: We associate \textit{Hanf numbers} \(\operatorname{H} (\mathbf{t})\) to triples \(\mathbf{t} = ( T , T_1 , p )\) where \(T\) and \(T_1\) are theories and \(p\) is a type. We show that the Hanf number for the property: ``there is a model \(M_1\) of \(T_1\) which omits \(p\), but \(M_1 \upharpoonright \tau\) is saturated'' is larger than the Hanf number of \(\mathcal{L}_{\lambda^+ , \kappa}\) but smaller than the Hanf number of \(\mathcal{L}_{( 2^\lambda )^+ , \kappa}\) when \(T\) is stable with \(\kappa = \kappa ( T )\). In fact, surprisingly, we even characterise the Hanf number of \(\mathbf{t}\) when we fix \(( T , \lambda )\) where \(T\) is a first order complete (and stable), \( \lambda \geq | T |\) and demand \(| T_1 | \leq \lambda \).
{{\copyright} 2020 Wiley-VCH GmbH}Weakly and locally positive Robinson theorieshttps://zbmath.org/1521.031122023-11-13T18:48:18.785376Z"Belkasmi, Mohammed"https://zbmath.org/authors/?q=ai:belkasmi.mohammedSummary: We introduce the notions of weakly and locally positive Robinson theories. We give a characterization of weakly positive Robinson theories by the amalgamation property, and a syntactic characterization of locally positive Robinson theories.
{{\copyright} 2021 Wiley-VCH GmbH}Maximality of linear continuous logichttps://zbmath.org/1521.031132023-11-13T18:48:18.785376Z"Malekghasemi, Mahya"https://zbmath.org/authors/?q=ai:malekghasemi.mahya"Bagheri, Seyed-mohammad"https://zbmath.org/authors/?q=ai:bagheri.seyed-mohammadSummary: The linear compactness theorem is a variant of the compactness theorem holding for linear formulas. We show that the linear fragment of continuous logic is maximal with respect to the linear compactness theorem and the linear elementary chain property. We also characterize linear formulas as those preserved by the ultramean construction.Topological elementary equivalence of regular semi-algebraic sets in three-dimensional spacehttps://zbmath.org/1521.031142023-11-13T18:48:18.785376Z"Geerts, Floris"https://zbmath.org/authors/?q=ai:geerts.floris"Kuijpers, Bart"https://zbmath.org/authors/?q=ai:kuijpers.bart-h-mSummary: We consider semi-algebraic sets and properties of these sets that are expressible by sentences in first-order logic over the reals. We are interested in first-order properties that are invariant under topological transformations of the ambient space. Two semi-algebraic sets are called topologically elementarily equivalent if they cannot be distinguished by such topological first-order sentences. So far, only semi-algebraic sets in one and two-dimensional space have been considered in this context. Our contribution is a natural characterisation of topological elementary equivalence of regular closed semi-algebraic sets in three-dimensional space, extending a known characterisation for the two-dimensional case. Our characterisation is based on the local topological behaviour of semi-algebraic sets and the key observation that topologically elementarily equivalent sets can be transformed into each other by means of geometric transformations, each of them mapping a set to a first-order indistinguishable one.Spectra and satisfiability for logics with successor and a unary functionhttps://zbmath.org/1521.031152023-11-13T18:48:18.785376Z"Milchior, Arthur"https://zbmath.org/authors/?q=ai:milchior.arthurSummary: We investigate the expressive power of two logics, both with the successor function: first-order logic with an uninterpreted function, and existential monadic second order logic -- that is first-order logic over words --, with multiplication by a constant \(b\). We prove that all \(b\)-recognizable sets are spectra of those logics. Furthermore, it is proven that some encoding of the set of halting times of a non-deterministic 2-counter automaton is also a spectrum. This yields undecidability of the finite satisfiability problem for those logics. Finally, it is shown that first-order logic with one uninterpreted function and successor can encode quickly increasing functions, such as the Knuth's up-arrows.Russell's typicality as another randomness notionhttps://zbmath.org/1521.031252023-11-13T18:48:18.785376Z"Tzouvaras, Athanassios"https://zbmath.org/authors/?q=ai:tzouvaras.athanassiosSummary: We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of ``small'' cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain \(M\) satisfies \(\operatorname{cf} ( | M | ) > \aleph_0\), then there exist \(| M |\) typical elements and only \(< | M |\) non-typical ones. In particular this is true for the standard model \(\mathcal{R}\) of second-order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to \(\subseteq )\) with the classes of ML-random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways).
{{\copyright} 2020 Wiley-VCH GmbH}The classification of countable models of set theoryhttps://zbmath.org/1521.031452023-11-13T18:48:18.785376Z"Clemens, John"https://zbmath.org/authors/?q=ai:clemens.john-daniel"Coskey, Samuel"https://zbmath.org/authors/?q=ai:coskey.samuel"Dworetzky, Samuel"https://zbmath.org/authors/?q=ai:dworetzky.samuelSummary: We study the complexity of the classification problem for countable models of set theory \((\mathsf{ZFC})\). We prove that the classification of arbitrary countable models of \(\mathsf{ZFC}\) is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of \(\mathsf{ZFC}\).Łoś's theorem and the axiom of choicehttps://zbmath.org/1521.031642023-11-13T18:48:18.785376Z"Tachtsis, Eleftherios"https://zbmath.org/authors/?q=ai:tachtsis.eleftheriosSummary: In set theory without the Axiom of Choice \((\mathsf{AC})\), we investigate the problem of the placement of Łoś's Theorem \((\mathsf{LT})\) in the hierarchy of weak choice principles, and answer several open questions from the book [Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)] by \textit{P. Howard} and \textit{J. E. Rubin}, as well as an open question by Brunner. We prove a number of results summarised in \S 3.On the relative strengths of fragments of collectionhttps://zbmath.org/1521.031682023-11-13T18:48:18.785376Z"McKenzie, Zachiri"https://zbmath.org/authors/?q=ai:mckenzie.zachiriSummary: Let \(\mathbf{M}\) be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, \(\Delta_0\)-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to \(\mathbf{M}\). We focus on two common parameterisations of collection: \(\Pi_n\)-\textit{collection}, which is the usual collection scheme restricted to \(\Pi_n\)-formulae, and \textit{strong \(\Pi_n\)-collection}, which is equivalent to \(\Pi_n\)-collection plus \(\Sigma_{n+1}\)-separation. The main result of this paper shows that for all \(n \geq 1\),
\begin{itemize}\item[1.] \(\mathbf{M}+\Pi_{n+1}\)-collection\(+\Sigma_{n+2}\)-induction on \(\omega\) proves the consistency of Zermelo Set Theory plus \(\Pi_{n}\)-collection,
\item[2.] the theory \(\mathbf{M}+\Pi_{n+1}\)-collection is \(\Pi_{n+3}\)-conservative over the theory \(\mathbf{M}+\) strong \(\Pi_n\)-collection.\end{itemize}
It is also shown that (2) holds for \(n=0\) when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and \(V=L\)) that does not include the powerset axiom.Choiceless large cardinals and set-theoretic potentialismhttps://zbmath.org/1521.031972023-11-13T18:48:18.785376Z"Cutolo, Raffaella"https://zbmath.org/authors/?q=ai:cutolo.raffaella"Hamkins, Joel David"https://zbmath.org/authors/?q=ai:hamkins.joel-davidSummary: We define a \textit{potentialist system} of \(\mathsf{ZF}\)-structures, i.e., a collection of \textit{possible worlds} in the language of \(\mathsf{ZF}\) connected by a binary \textit{accessibility relation}, achieving a potentialist account of the full background set-theoretic universe \(V\). The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just \(\mathsf{ZF}\). It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory \(\mathsf{S4.2}\). Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory \(\mathsf{S5}\), both for assertions in the language of \(\mathsf{ZF}\) and for assertions in the full potentialist language.
{{\copyright} 2022 Wiley-VCH GmbH.}Bounded finite set theoryhttps://zbmath.org/1521.032082023-11-13T18:48:18.785376Z"Kirby, Laurence"https://zbmath.org/authors/?q=ai:kirby.laurenceSummary: We define an axiom schema \(\mathsf{I} \Delta_0 \mathsf{S}\) for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, \( \mathsf{I} \Delta_0\), and use some of its basic model theory to establish some independence results for various axioms of set theory over \(\mathsf{I} \Delta_0 \mathsf{S} \). Then we ask: given a model \(M\) of \(\mathsf{I} \Delta_0\), is there a model of \(\mathsf{I} \Delta_0 \mathsf{S}\) whose ordinal arithmetic is isomorphic to \(M\)? We show that the answer is yes if \(M \models \mathsf{Exp} \).
{{\copyright} 2021 Wiley-VCH GmbH}Indivisible sets and well-founded orientations of the Rado graphhttps://zbmath.org/1521.032142023-11-13T18:48:18.785376Z"Ackerman, Nathanael L."https://zbmath.org/authors/?q=ai:ackerman.nathanael-leedom"Brian, Will"https://zbmath.org/authors/?q=ai:brian.william-reaSummary: Every set can been thought of as a directed graph whose edge relation is \(\varepsilon\). We show that many natural examples of directed graphs of this kind are indivisible: \(\mathbf{H}_\kappa\) for every infinite \(\kappa\), \(\mathbf{V}_\lambda\) for every indecomposable \(\lambda\), and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find \(2^{\aleph_0}\) indivisible well-founded orientations of the random graph that are distinct up to isomorphism, and \(\aleph_1\) that are distinct up to siblinghood.Parametric Presburger arithmetic: complexity of counting and quantifier eliminationhttps://zbmath.org/1521.032222023-11-13T18:48:18.785376Z"Bogart, Tristram"https://zbmath.org/authors/?q=ai:bogart.tristram"Goodrick, John"https://zbmath.org/authors/?q=ai:goodrick.john"Nguyen, Danny"https://zbmath.org/authors/?q=ai:nguyen.danny"Woods, Kevin"https://zbmath.org/authors/?q=ai:woods.kevin-m|woods.kevinSummary: We consider an expansion of Presburger arithmetic which allows multiplication by \(k\) parameters \(t_1,\dots,t_k\). A formula in this language defines a parametric set \(S_{\mathbf{t}} \subseteq\mathbb{Z}^d\) as \(\mathbf{t}\) varies in \(\mathbb{Z}^k\), and we examine the counting function \(|S_{\mathbf{t}}|\) as a function of \(\mathbf{t}\). For a single parameter, it is known that \(|S_t|\) can be expressed as an eventual quasi-polynomial (there is a period \(m\) such that, for sufficiently large \(t\), the function is polynomial on each of the residue classes mod \(m)\). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming \(\mathbf{P}\neq\mathbf{NP})\) we construct a parametric set \(S_{t_1,t_2}\) such that \(|S_{t_1,t_2}|\) is not even polynomial-time computable on input \((t_1,t_2)\). In contrast, for parametric sets \(S_{\mathbf{t}}\subseteq\mathbb{Z}^d\) with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that \(| S_{\mathbf{t}}|\) is always polynomial-time computable in the size of \(\mathbf{t}\), and in fact can be represented using the gcd and similar functions.Rigid models of Presburger arithmetichttps://zbmath.org/1521.032252023-11-13T18:48:18.785376Z"Jeřábek, Emil"https://zbmath.org/authors/?q=ai:jerabek.emilSummary: We present a description of rigid models of Presburger arithmetic (i.e., \(\mathbb{Z}\)-groups). In particular, we show that Presburger arithmetic has rigid models of all infinite cardinalities up to the continuum, but no larger.Models of relevant arithmetichttps://zbmath.org/1521.032322023-11-13T18:48:18.785376Z"Slaney, John"https://zbmath.org/authors/?q=ai:slaney.john-kSummary: It is well known that the relevant arithmetic \(\mathbf{R}^\#\) admits finite models whose domains are the integers \textit{modulo}~\(n\) rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which \(\mathbf{R}^\#\) can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic \(\mathbf{RM}^\#\) \textit{modulo}~\(n\) and a partial account for the case of \(\mathbf{R}^\#\) \textit{modulo} a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that `\(0=1\)' implies everything.Consistency and decidability in some paraconsistent arithmeticshttps://zbmath.org/1521.032332023-11-13T18:48:18.785376Z"Tedder, Andrew"https://zbmath.org/authors/?q=ai:tedder.andrewSummary: The standard style of argument used to prove that a theory is undecidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent setting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in inconsistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals.The small-is-very-small principlehttps://zbmath.org/1521.032342023-11-13T18:48:18.785376Z"Visser, Albert"https://zbmath.org/authors/?q=ai:visser.albertSummary: The central result of this paper is \textit{the small-is-very-small principle} for restricted sequential theories. The principle says roughly that whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness: i.e., a witness below a given standard number. Which cuts are sufficiently small will depend on the complexity of the formula defining the property. We draw various consequences from the central result. E.g., roughly speaking, (i) every restricted, recursively enumerable sequential theory has a finitely axiomatized extension that is conservative with respect to formulas of complexity \(\leq n\); (ii) every sequential model has, for any \(n\), an extension that is elementary for formulas of complexity \(\leq n\), in which the intersection of all definable cuts is the natural numbers; (iii) we have reflection for \(\Sigma_2^0\)-sentences with sufficiently small witness in any consistent restricted theory \(U\); (iv) suppose \(U\) is recursively enumerable and sequential. Suppose further that every recursively enumerable and sequential \(V\) that locally inteprets \(U\), globally interprets \(U\). Then, \(U\) is mutually globally interpretable with a finitely axiomatized sequential theory. The paper contains some careful groundwork developing partial satisfaction predicates in sequential theories for the complexity measure \textit{depth of quantifier alternations}.Sheaves of structures, Heyting-valued structures, and a generalization of Łoś's theoremhttps://zbmath.org/1521.032562023-11-13T18:48:18.785376Z"Aratake, Hisashi"https://zbmath.org/authors/?q=ai:aratake.hisashiSummary: Sheaves of structures are useful to give constructions in universal algebra and model theory. We can describe their logical behavior in terms of Heyting-valued structures. In this paper, we first provide a systematic treatment of sheaves of structures and Heyting-valued structures from the viewpoint of categorical logic. We then prove a form of Łoś's theorem for Heyting-valued structures. We also give a characterization of Heyting-valued structures for which Łoś's theorem holds with respect to any maximal filter.
{{\copyright} 2021 Wiley-VCH GmbH}Nonstandard methods for finite structureshttps://zbmath.org/1521.032582023-11-13T18:48:18.785376Z"Tsuboi, Akito"https://zbmath.org/authors/?q=ai:tsuboi.akitoSummary: We discuss the possibility of applying the compactness theorem to the study of finite structures. Given a class of finite structures, it is important to determine whether it can be expressed by a particular category of sentences. We are interested in this type of problem, and use nonstandard method for showing the non-expressibility of certain classes of finite graphs by an existential monadic second order sentence.
{{\copyright} 2020 Wiley-VCH GmbH}Which subsets of an infinite random graph look random?https://zbmath.org/1521.051802023-11-13T18:48:18.785376Z"Brian, Will"https://zbmath.org/authors/?q=ai:brian.william-reaSummary: Given a countable graph, we say a set \(A\) of its vertices is universal if it contains every countable graph as an induced subgraph, and \(A\) is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on \(\mathbb{N}\), (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on \(\mathbb{N}\) will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.Evolving Shelah-Spencer graphshttps://zbmath.org/1521.051822023-11-13T18:48:18.785376Z"Elwes, Richard"https://zbmath.org/authors/?q=ai:elwes.richard-hSummary: We define an \textit{evolving Shelah-Spencer process} as one by which a random graph grows, with at each time \(\tau \in \mathbb{N}\) a new node incorporated and attached to each previous node with probability \(\tau^{- \alpha}\), where \(\alpha \in ( 0 , 1 ) \setminus \mathbb{Q}\) is fixed. We analyse the graphs that result from this process, including the infinite limit, in comparison to Shelah-Spencer sparse random graphs discussed in [\textit{J. Spencer}, The strange logic of random graphs. Berlin: Springer (2001; Zbl 0976.05001)] and throughout the model-theoretic literature. The first order axiomatisation for classical Shelah-Spencer graphs comprises a \textit{Generic Extension} axiom scheme and a \textit{No Dense Subgraphs} axiom scheme. We show that in our context \textit{Generic Extension} continues to hold. While \textit{No Dense Subgraphs} fails, a weaker \textit{Few Rigid Subgraphs} property holds.
{{\copyright} 2021 Wiley-VCH GmbH}Surjectively rigid chainshttps://zbmath.org/1521.060012023-11-13T18:48:18.785376Z"Montalvo-Ballesteros, Mayra"https://zbmath.org/authors/?q=ai:montalvo-ballesteros.mayra"Truss, John K."https://zbmath.org/authors/?q=ai:truss.john-kennethSummary: We study rigidity properties of linearly ordered sets (chains) under automorphisms, embeddings, epimorphisms, and endomorphisms. We focus on two main cases: dense subchains of the real numbers, and uncountable dense chains of higher regular cardinalities. We also give a Fraenkel-Mostowski model which illustrates the role of the axiom of choice in one of the key proofs.
{{\copyright} 2021 Wiley-VCH GmbH}Model completion of scaled lattices and co-Heyting algebras of \(p\)-adic semi-algebraic setshttps://zbmath.org/1521.060032023-11-13T18:48:18.785376Z"Darnière, Luck"https://zbmath.org/authors/?q=ai:darniere.luckSummary: Let \(p\) be prime number, \(K\) be a \(p\)-adically closed field, \(X\subseteq K^m\) a semi-algebraic set defined over \(K\) and \(L(X)\) the lattice of semi-algebraic subsets of \(X\) which are closed in \(X\). We prove that the complete theory of \(L(X)\) eliminates quantifiers in a certain language \(\mathcal{L}_{\operatorname{ASC}}\), the \(\mathcal{L}_{\operatorname{ASC}}\)-structure on \(L(X)\) being an extension by definition of the lattice structure. Moreover it is decidable, contrary to what happens over a real closed field for \(m>1\). We classify these \(\mathcal{L}_{\operatorname{ASC}}\)-structures up to elementary equivalence, and get in particular that the complete theory of \(L(K^m)\) only depends on \(m\), not on \(K\) nor even on \(p\). As an application we obtain a classification of semi-algebraic sets over countable \(p\)-adically closed fields up to so-called ``pre-algebraic'' homeomorphisms.Ordered fields dense in their real closure and definable convex valuationshttps://zbmath.org/1521.120092023-11-13T18:48:18.785376Z"Krapp, Lothar Sebastian"https://zbmath.org/authors/?q=ai:krapp.lothar-sebastian"Kuhlmann, Salma"https://zbmath.org/authors/?q=ai:kuhlmann.salma"Lehéricy, Gabriel"https://zbmath.org/authors/?q=ai:lehericy.gabrielRecall that a theorem of Artin and Schreier states that every ordered field \(K\) admits algebraic extension, called its \textit{real closure}, which is a real closed field whose ordering extends the ordering of \(K\) and is unique up to a unique isomorphism of fields over \(K\). In this article, as its title says, the authors study the class of ordered fields which are dense in their real closure. Very much in the same spirit, the authors also study the class of ordered abelian groups which are dense in their divisible hull. Their main results include characterizations and axiomatizations of such classes by generally describing classes of ordered structures which are dense in certain of their definable closures (see Theorem 4.1 for ordered fields and Corollary 3.3 for ordered abelian groups).
Their study is motivated by the following recurrent model theoretic question: which Henselian valuations are \textit{definable} in the language of rings (resp. ordered rings)? Their results on the above-mentioned classes of structures helped them to prove the following result (Theorem 5.3):
Theorem: Let \((K,<)\) be an ordered field and \(v\) a Henselian valuation on \(K\). Suppose that at least one of the following holds.
\begin{itemize}
\item[1.] \(vK\) (the value group) is discretely ordered.
\item[2.] \(vK\) has a limit point in the divisible hull of \(vK\) over \(vK\).
\item[3.] \(Kv\) (the residue field) has a limit point in the real closure of \(Kv\) over \(Kv\).
\end{itemize}
Then \(v\) is definable in \(K\) in the language of ordered rings. Moreover, in the cases (1) and (2), \(v\) is definable by a formula in the language of ordered rings with one parameter.
Reviewer: Pablo Cubides Kovacsics (Bogotá)Elimination of imaginaries in \(\mathbb{C}((\Gamma))\)https://zbmath.org/1521.120112023-11-13T18:48:18.785376Z"Vicaría, Mariana"https://zbmath.org/authors/?q=ai:vicaria.marianaSummary: In this paper, we study elimination of imaginaries in henselian valued fields of equicharacteristic zero and residue field algebraically closed. The results are sensitive to the complexity of the value group. We focus first on the case where the ordered abelian group has finite spines, and then prove a better result for the dp-minimal case. In previous work the author proved that an ordered abelian with finite spines weakly eliminates imaginaries once one adds sorts for the quotient groups \(\Gamma / \Delta\) for each definable convex subgroup \(\Delta\), and sorts for the quotient groups \(\Gamma /(\Delta + l\Gamma)\) where \(\Delta\) is a definable convex subgroup and \(l \in \mathbb{N}_{\ge 2}\). We refer to these sorts as the \textit{quotient sorts}. Jahnke, Simon, and Walsberg [ \textit{F. Jahnke} et al., J. Symb. Log. 82, No. 1, 151--165 (2017; Zbl 1385.03040)] characterized \(dp\)-minimal ordered abelian groups as those without singular primes, that is, for every prime number \(p\) one has \([\Gamma :p\Gamma ]< \infty\).
We prove the following two theorems:
\textbf{Theorem} \textit{Let \(K\) be a henselian valued field of equicharacteristic zero with residue field algebraically closed and value group of finite spines. Then \(K\) admits weak elimination of imaginaries once one adds codes for all the definable \(\mathcal{O}\)-submodules of \(K^n\) for each \(n \in \mathbb{N}\), and the quotient sorts for the value group.}
\textbf{Theorem} \textit{Let \(K\) be a henselian valued field of equicharacteristic zero, with residue field algebraically closed and whose value group is dp-minimal. Then \(K\) eliminates imaginaries once one adds codes for all the definable \(\mathcal{O}\)-submodules of \(K^n\) for each \(n \in \mathbb{N}\), the quotient sorts for the value group and constants to distinguish the elements of each of the finite groups \(\Gamma /\ell \Gamma\), where \(\ell \in \mathbb{N}_{>0}\).}Universal cohomology theorieshttps://zbmath.org/1521.180052023-11-13T18:48:18.785376Z"Barbieri-Viale, Luca"https://zbmath.org/authors/?q=ai:barbieri-viale.lucaThe axiomatic approach to homology theories, in particular to singular versus cellular homology, as it was introduced by \textit{S. Eilenberg} and \textit{N. E. Steenrod} [Proc. Natl. Acad. Sci. USA 31, 117--120 (1945; Zbl 0061.40504); Foundations of algebraic topology. Princeton, NJ: Princeton University Press (1952; Zbl 0047.41402)] in topology, has been largely influential and their unicity theorem quite astonishing. The first key step in this story was taken around the years 1945--1952, while a ramified study of topological generalized (co)homology theories emerged for a start [\textit{E. Dyer}, Cohomology theories. Reading, MA: The Benjamin/Cummings Publishing Company (1969; Zbl 0182.57002)].
A parallel history can be seen in Grothendieck construction of a Weil cohomology in algebraic geometry, which was started from a wish-list of axioms and was afforded in the years 1958--1966 after two other key foundational steps, the first step being in homological algebra with the concept of satellite and that of \(\partial\)-functor [\textit{H. Cartan} and \textit{S. Eilenberg}, Homological algebra. Princeton, NJ: Princeton University Press (1956; Zbl 0075.24305); \textit{A. Grothendieck}, Tôhoku Math. J. (2) 9, 119--221 (1957; Zbl 0118.26104)] while the second step being an introduction of Grothendieck topologies. Notably, the stride from Weil cohomology to Grothendieck motives and motivic cohomology was originally based on the third foundational step of the Tannakian formalism, which is still dependent on the standard conjectures [\textit{P. Deligne} and \textit{J. S. Milne}, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. Broadly speaking, the category of motives in Grothendieck vision can be regarded as a way to express a sort of abelian envelope of algebraic varieties, while motivic cohomology can be the abelian avatar of a variety.
\textit{P. Freyd} [Proc. Conf. Categor. Algebra, La Jolla 1965, 95--120 (1966; Zbl 0202.32402)] considered the question how nicely a given category can be represented in an abelian category, though his universal abelian category of an additive category has not been linked to the construction of motives. Freyd also observed that there is an embedding of a triangulated category in an abelian category which is universal with respect to homological functors, while \textit{A. Heller} [Bull. Am. Math. Soc. 74, 28--63 (1968; Zbl 0177.25605)] constructed a universal homology in a stable abelian category.
Actually, on the algebraic side of the story, around 1997, Nori provided a universal abelian category, making use of a variant of the Tannakian formalism [\textit{A. Huber} and \textit{S. Müller-Stach}, Periods and Nori motives. Cham: Springer (2017; Zbl 1369.14001)], while \textit{P. Deligne} [in: Galois groups over \(\mathbb{Q}\), Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, . 79--297 (1989; Zbl 0742.14022)] introduced the abelian category of mixed realizations and \textit{Y. André} [Publ. Math., Inst. Hautes Étud. Sci. 83, 5--49 (1996; Zbl 0874.14010)] proposed motivated cycles showing how to shirk the standard conjectures. Nontheless, Nori's idea as well as André's and Deligne's is based on the Tannakian formalism, making use of existing fiber functors, so that the standard conjectures remain unsolved. The approach due to \textit{V. Voevodsky} [Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)] and \textit{J. Ayoub} [J. Reine Angew. Math. 693, 1--149 (2014; Zbl 1299.14020)] gives the construction of a triangulated category of motives in place of the abelian category, though the correct motivic t-structure is missing.
The unified (co)homological framework in [\textit{L. Barbieri-Viale}, J. Pure Appl. Algebra 221, No. 7, 1565--1588 (2017; Zbl 1375.14019)] is also a first step toward the construction of motives independently of fiber functors, being settled in the language of categorical model theory. Its translation in the language of representations of quivers started in parallel with [\textit{L. Barbieri-Viale} and \textit{M. Prest}, Rend. Semin. Mat. Univ. Padova 139, 205--224 (2018; Zbl 1395.18012)]. Actually, Freyd's universal abelian category is linked with the construction of Nori motives as well as with the triangulated categories of Voevodsky motives. A tensor version of Freyd's universal abelian category provides tensor product of motives, e.g. for abelian categories modeles on a given cohomology satisfying Künnetuh formula [\textit{L. Barbieri-Viale} et al., Pac. J. Math. 306, No. 1, 1--30 (2020; Zbl 1454.14062); \textit{L. Barbieri-Viale} and \textit{M. Prest}, J. Pure Appl. Algebra 224, No. 6, Article ID 106267, 13 p. (2020; Zbl 1439.14077)].
The principal objective in this paper is to show that there is a simple algebraic picture providing universal (co)homology theories in abelian categories, independent of the Tannakian formalism, revisiting and developing the previously hinted construction of theoretical motives [\textit{L. Barbieri-Viale}, J. Pure Appl. Algebra 221, No. 7, 1565--1588 (2017; Zbl 1375.14019)]. This unified framework for (co)homology theories on any fixed category \(\mathcal{C}\) with values in variable abelian categories \(\mathcal{A}\) is achieved through the solution of representability problems.
Reviewer: Hirokazu Nishimura (Tsukuba)Tameness in generalized metric structureshttps://zbmath.org/1521.180082023-11-13T18:48:18.785376Z"Lieberman, Michael"https://zbmath.org/authors/?q=ai:lieberman.michael-j"Rosický, Jiří"https://zbmath.org/authors/?q=ai:rosicky.jiri"Zambrano, Pedro"https://zbmath.org/authors/?q=ai:zambrano.pedroThe authors introduce \(\mathbb{V}\)-abstract elementary classes (\(\mathbb{V}\)-AEC's), generalizing the notion of metric abstract elementary class, introduced in [\textit{M. Lieberman} and \textit{J. Rosický}, J. Symb. Log. 82, No. 3, 1022--1040 (2017; Zbl 1422.03084)]. Here the notion of distance between two elements of a structure \(\mathcal{M}\) is given by an element of what the authors call a Flagg quantale \(\mathbb{V}\), rather than by a real number, and accordingly \(k\)-ary relational and function symbols of a given signature are interpreted as non-expanding maps \(R^{\mathcal{M}} \colon M^k \to \mathbb{V}\) and \(f^{\mathcal{M}} \colon M^k \to M,\) respectively.
Their main result (Theorem 6.7) states that, under the assumption of existence of a suitable compact cardinal, every pseudo-\(\mathbb{V}\)-AEC is \(\mathbb{V}\)-tame. Tameness is a condition that has to do with the existence of a cardinal so that distinguishable (say, via automorphisms) elements of a model \(M\) can already be distinguished within a submodel of at most that cardinality. The \(\mathbb{V}\)-version of the notion naturally incorporates the underlying \(\mathbb{V}\)-valued distance of elements into the idea of being (in)distinguishable. The prefix pseudo- has to do with considering a quantale-valued distance as a function \(d \colon M \times M \to \mathbb{V}\) satisfying the expected properties except that \(d(x,y)=0\) implies \(x=y.\)
Apart from presenting the above result and the necessary machinery leading to it, the introductory section of the paper on quantales provides valuable information and a host of interesting examples that offer sufficient motivation for the extra step in generalization. The idea of dropping the requirement that \(d(x,x)=0\) is also discussed in connection to the perspective of treating sheaves as \(\mathbb{V}\)-valued sets, in case our quantale is simply a frame.
Two references should be corrected, [28] to [\textit{D. Hofmann} and \textit{C. D. Reis}, Categ. Gen. Algebr. Struct. Appl. 9, No. 1, 77--138 (2018; Zbl 1407.18002)] and [41] to [\textit{C. J. Mulvey}, Suppl. Rend. Circ. Mat. Palermo (2) 12, 99--104 (1986; Zbl 0633.46065)].
Reviewer: Panagis Karazeris (Pátra)Unipotence in positive characteristic for groups of finite Morley rankhttps://zbmath.org/1521.200682023-11-13T18:48:18.785376Z"Tindzogho Ntsiri, Jules Gael"https://zbmath.org/authors/?q=ai:tindzogho-ntsiri.jules-gaelIn the paper under review, the author defines a new form of unipotence in groups of finite Morley rank which extends \(0\)-unipotence (as defined by \textit{J. Burdges} in [J. Algebra 274, No. 1, 215--229 (2004; Zbl 1053.20029)]) to any characteristic.
The main results proved in this paper are the following:
Theorem 5.12: Let \(G\) be a connected nilpotent group of finite Morley rank. Then \(G/Z(G)\) is a unipotent group of finite Morley rank.
Theorem 5.13: Let \(G\) be a connected solvable group of finite Morley rank. There exists a pair \((H,\mathbf{T})\) of connected definable subgroups such that
(i) \(G=H\ast \mathbf{T}\);
(ii) \(\mathbf{T}\) is a generalised pseudotorus;
(iii) \(H'\) is a good unipotent subgroup of finite Morley rank.
Reviewer: Enrico Jabara (Venezia)Peterzil-Steinhorn subgroups and \(\mu\)-stabilizers in ACFhttps://zbmath.org/1521.201082023-11-13T18:48:18.785376Z"Kamensky, Moshe"https://zbmath.org/authors/?q=ai:kamensky.moshe"Starchenko, Sergei"https://zbmath.org/authors/?q=ai:starchenko.sergei"Ye, Jinhe"https://zbmath.org/authors/?q=ai:ye.jinheSummary: We consider \(G\), a linear algebraic group defined over \(\Bbbk\), an algebraically closed field (ACF). By considering \(\Bbbk\) as an embedded residue field of an algebraically closed valued field \(K\), we can associate to it a compact \(G\)-space \(S^\mu_G(\Bbbk)\) consisting of \(\mu\)-types on \(G\). We show that for each \(p_\mu \in S^\mu_G(\Bbbk)\), \(\mathrm{Stab}^\mu (p) = \mathrm{Stab}(p_\mu)\) is a solvable infinite algebraic group when \(p_\mu\) is centered at infinity and residually algebraic. Moreover, we give a description of the dimension of \(\mathrm{Stab}(p_\mu)\) in terms of the dimension of \(p\).Defining \(R\) and \(G(R)\)https://zbmath.org/1521.201112023-11-13T18:48:18.785376Z"Segal, Dan"https://zbmath.org/authors/?q=ai:segal.dan.1|segal.dan"Tent, Katrin"https://zbmath.org/authors/?q=ai:tent.katrinSummary: We show that for Chevalley groups \(G ( R )\) of rank at least 2 over an integral domain \(R\) each root subgroup is (essentially) the double centralizer of a corresponding root element. In many cases, this implies that \(R\) and \(G ( R )\) are bi-interpretable, yielding a new approach to bi-interpretability for algebraic groups over a wide range of rings and fields.
For such groups it then follows that the group \(G ( R )\) is (finitely) axiomatizable in the appropriate class of groups provided \(R\) is (finitely) axiomatizable in the corresponding class of rings.Malgrange division by quasianalytic functionshttps://zbmath.org/1521.320082023-11-13T18:48:18.785376Z"Bierstone, Edward"https://zbmath.org/authors/?q=ai:bierstone.edward"Milman, Pierre D."https://zbmath.org/authors/?q=ai:milman.pierre-dSummary: Quasianalytic classes are classes of \(\mathcal{C}^\infty\) functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of \(\mathcal{C}^\infty\) functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of \(\mathcal{C}^\infty\) by real-analytic functions.Verified quadratic virtual substitution for real arithmetichttps://zbmath.org/1521.682462023-11-13T18:48:18.785376Z"Scharager, Matias"https://zbmath.org/authors/?q=ai:scharager.matias"Cordwell, Katherine"https://zbmath.org/authors/?q=ai:cordwell.katherine"Mitsch, Stefan"https://zbmath.org/authors/?q=ai:mitsch.stefan"Platzer, André"https://zbmath.org/authors/?q=ai:platzer.andreSummary: This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the first-order logic of real arithmetic is decidable by QE. However, in practice, QE algorithms are highly complicated and often combine multiple methods for performance. VS is a practically successful method for QE that targets formulas with low-degree polynomials. To our knowledge, this is the first work to formalize VS for quadratic real arithmetic including inequalities. The proofs necessitate various contributions to the existing multivariate polynomial libraries in Isabelle/HOL. Our framework is modularized and easily expandable (to facilitate integrating future optimizations), and could serve as a basis for developing practical general-purpose QE algorithms. Further, as our formalization is designed with practicality in mind, we export our development to SML and test the resulting code on 378 benchmarks from the literature, comparing to Redlog, Z3, Wolfram Engine, and SMT-RAT. This identified inconsistencies in some tools, underscoring the significance of a verified approach for the intricacies of real arithmetic.
For the entire collection see [Zbl 1509.68013].An introduction to gravitational waves through electrodynamics: a quadrupole comparisonhttps://zbmath.org/1521.830242023-11-13T18:48:18.785376Z"Dorsch, Glauber Carvalho"https://zbmath.org/authors/?q=ai:dorsch.glauber-carvalho"Antunes Porto, Lucas Emanuel"https://zbmath.org/authors/?q=ai:antunes-porto.lucas-emanuel(no abstract)Kähler-Einstein metrics near an isolated log-canonical singularityhttps://zbmath.org/1521.831642023-11-13T18:48:18.785376Z"Datar, Ved"https://zbmath.org/authors/?q=ai:datar.ved-v"Fu, Xin"https://zbmath.org/authors/?q=ai:fu.xin.1"Song, Jian"https://zbmath.org/authors/?q=ai:song.jianSummary: We construct Kähler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler-Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler-Einstein metrics from hyperbolic geometry.