Recent zbMATH articles in MSC 03Chttps://zbmath.org/atom/cc/03C2024-07-25T18:28:20.333415ZUnknown authorWerkzeugTeichmüller theory: classical, higher, super and quantum. Abstracts from the workshop held July 30 -- August 4, 2023https://zbmath.org/1537.000282024-07-25T18:28:20.333415ZSummary: Teichmüller spaces play a major role in many areas of mathematics and physical science. The subject of the conference was recent developments of Teichmüller theory with its different ramifications that include the classical, the higher, the super and the quantum aspects of the theory.On the local coordination of fuzzy valuationshttps://zbmath.org/1537.030222024-07-25T18:28:20.333415Z"Yakh''yaeva, Gul'nara Èrkinovna"https://zbmath.org/authors/?q=ai:yakhyaeva.gulnara-erkinovnaSummary: The paper is devoted to the model-theoretic formalization of the semantic model of the object domain. The article discusses the concept of a fuzzy model, which is a model where the truth function exhibits properties of a fuzzy measure. We demonstrate that a fuzzy model is a generalization of the concept of fuzzification of a precedent (semantic) model to include a countable number of precedents. In practice, it is common to have partial expert knowledge about the set of events in the object domain, making it difficult to immediately describe the fuzzy model. Additionally, since expert valuations are subjective, they may be incorrect and inconsistent with any fuzzy model. In the article, we introduce the concepts of coordinated and locally coordinated valuation of a set of sentences, and provide proofs for interval theorems and an analogue of the compactness theorem.Classification of \(\aleph_0\)-categorical \(C\)-minimal pure \(C\)-setshttps://zbmath.org/1537.030302024-07-25T18:28:20.333415Z"Delon, Françoise"https://zbmath.org/authors/?q=ai:delon.francoise"Mourgues, Marie-Hélène"https://zbmath.org/authors/?q=ai:mourgues.marie-heleneThe present paper concerns studying countably categorical and \(C\)-minimal pure \(C\)-structures (structures equipped with a ternary \(C\)-relation). The authors introduce the notions of a solvable good tree, a colored good tree and a precolored good tree. They prove that a pure \(C\)-set \(M\) is indiscernible, finite or countably categorical and \(C\)-minimal iff its canonical tree \(T(M)\) is a colored good tree. The main result of the paper is a classification of all countably categorical and \(C\)-minimal \(C\)-sets up to elementary equivalence.
Reviewer: Beibut Kulpeshov (Almaty)Canonical functions: a proof via topological dynamicshttps://zbmath.org/1537.030312024-07-25T18:28:20.333415Z"Pinsker, Michael"https://zbmath.org/authors/?q=ai:pinsker.michael"Bodirsky, Manuel"https://zbmath.org/authors/?q=ai:bodirsky.manuelSummary: Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity.Two-variable logic has weak, but not strong, Beth definabilityhttps://zbmath.org/1537.030322024-07-25T18:28:20.333415Z"Andréka, Hajnal"https://zbmath.org/authors/?q=ai:andreka.hajnal"Németi, István"https://zbmath.org/authors/?q=ai:nemeti.istvanSummary: We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.Forking, imaginaries, and other features of ACFGhttps://zbmath.org/1537.030332024-07-25T18:28:20.333415Z"D'Elbée, Christian"https://zbmath.org/authors/?q=ai:delbee.christianSummary: We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called ACFG. This theory was introduced in [the author, J. Math. Log. 21, No. 3, Article ID 2150016, 44 p. (2021; Zbl 07419666)] as a new example of \(\text{NSOP}_1\) nonsimple theory. In this paper we describe more features of ACFG, such as imaginaries. We also study various independence relations in ACFG, such as Kim-independence or forking independence, and describe interactions between them.The relativized Lascar groups, type-amalgamation, and algebraicityhttps://zbmath.org/1537.030342024-07-25T18:28:20.333415Z"Dobrowolski, Jan"https://zbmath.org/authors/?q=ai:dobrowolski.jan-cz"Kim, Byunghan"https://zbmath.org/authors/?q=ai:kim.byunghan"Kolesnikov, Alexei"https://zbmath.org/authors/?q=ai:kolesnikov.alexei-s"Lee, Junguk"https://zbmath.org/authors/?q=ai:lee.jungukSummary: In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory \(T\) is \(G\)-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if \(c\) is a finite tuple algebraic over a tuple \(a\), the Lascar group of \(\mathrm{stp}(ac)\) is abelian, and the underlying theory is \(G\)-compact, then the Lascar groups of \(\mathrm{stp}(ac)\) and of \(\mathrm{stp}(a)\) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of \(\mathrm{stp}(ac)\) is abelian, nor the assumption of \(c\) being finite can be removed.The degree of nonminimality is at most 2https://zbmath.org/1537.030352024-07-25T18:28:20.333415Z"Freitag, James"https://zbmath.org/authors/?q=ai:freitag.james"Jaoui, Rémi"https://zbmath.org/authors/?q=ai:jaoui.remi"Moosa, Rahim"https://zbmath.org/authors/?q=ai:moosa.rahim-nIn this short paper, which is extremely well-written, the authors show that for a stationary type of Lascar rank at least 2 in the theory DCF\(_{0,m}\) of differentially closed fields of characteristic \(0\) with \(m\) commuting derivations, its degree of non-minimality is at most \(2\). For DCF\(_{0,1}\), if the type is defined over the field of constants (that is, over the elements of derivation \(0\)), then the degree of non-minimality is \(1\).
Recall that Lascar rank (or \(U\)-rank) is the rank associated to the forking relation in superstable theories. If a stationary type \(p\) over \(A\) has a forking extension \(q\) over a superset \(B\supset A\), then so it does over a finite fragment of a Morley sequence of \(q\), by superstability, for the canonical base is always algebraic over finitely many elements of the Morley sequence. In particular, there are finitely many realizations \(c_i\) of \(q\) (and thus of \(p\)) such that \(p\) admits a forking extension over \(A\cup\{c_1,\ldots, c_d\}\). If \(p\) has rank at least \(2\), there is such a forking extension which is not algebraic. The degree of non-minimality of a stationary type of rank at least \(2\) is the least value \(d\) as above.
For the particular case of DCF\(_{0,1}\), in order to obtain the improved bound of \(\mathrm{nmdeg}(p)\le 1\), the authors use that for a stationary type defined over the constants which is internal to the constants, its binding group cannot be centerless.
Reviewer: Amador Martin-Pizarro (Freiburg)A strong failure of \(\aleph _0\)-stability for atomic classeshttps://zbmath.org/1537.030362024-07-25T18:28:20.333415Z"Laskowski, Michael C."https://zbmath.org/authors/?q=ai:laskowski.michael-chris"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: We study classes of atomic models \(\mathbf{At}_T\) of a countable, complete first-order theory \(T\). We prove that if \(\mathbf{At}_T\) is not \(\text{pcl}\)-small, i.e., there is an atomic model \(N\) that realizes uncountably many types over \(\text{pcl}_N(\bar{a})\) for some finite \(\bar{a}\) from \(N\), then there are \(2^{\aleph _1}\) non-isomorphic atomic models of \(T\), each of size \(\aleph _1\).Keisler's order is not simple (and simple theories may not be either)https://zbmath.org/1537.030372024-07-25T18:28:20.333415Z"Malliaris, M."https://zbmath.org/authors/?q=ai:malliaris.maryanthe-elizabeth"Shelah, S."https://zbmath.org/authors/?q=ai:shelah.saharonSummary: Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. In fact, it embeds \(\mathcal{P}(\omega) / \operatorname{fin} \). The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought of as densities of certain regular pairs, in the sense of Szemerédi's regularity lemma. The proof involves ideas from model theory, set theory, and finite combinatorics.HKSS-completeness of modal algebrashttps://zbmath.org/1537.030382024-07-25T18:28:20.333415Z"Bazhenov, Nikolay"https://zbmath.org/authors/?q=ai:bazhenov.n-aSummary: The paper studies computability-theoretic properties of countable modal algebras. We prove that the class of modal algebras is complete in the sense of the work of \textit{D. R. Hirschfeldt} et al. [Ann. Pure Appl. Logic 115, No. 1--3, 71--113 (2002; Zbl 1016.03034)]. This answers an open question of \textit{N. Bazhenov} [Stud. Log. 104, No. 6, 1083--1097 (2016; Zbl 1417.03233)]. The result implies that every degree spectrum and every categoricity spectrum can be realized by a suitable modal algebra.Structural highness notionshttps://zbmath.org/1537.030392024-07-25T18:28:20.333415Z"Calvert, Wesley"https://zbmath.org/authors/?q=ai:calvert.wesley"Franklin, Johanna N. Y."https://zbmath.org/authors/?q=ai:franklin.johanna-n-y"Turetsky, Dan"https://zbmath.org/authors/?q=ai:turetsky.danThe authors introduce and study several properties of highness for Turing degrees, like highness for isomorphism (a degree is high for isomorphism if for any two computable classically isomorphic structures it computes some isomorphism between them), uniform highness for isomorphism (the index of such an isomorphism with respect to this degree is computed uniformly from indices of these structures), (uniform) highness for paths, (uniform) highness for isomorphism for Harrison orders, highness for (tight) descending sequences, and also the notions of Scott completeness and jump completeness.
The relationship between these properties and other known properties of degrees are also studied.
Reviewer: Andrei S. Morozov (Novosibirsk)Punctually presented structures. I: Closure theoremshttps://zbmath.org/1537.030402024-07-25T18:28:20.333415Z"Dorzhieva, Marina"https://zbmath.org/authors/?q=ai:dorzhieva.marina-valerianovna"Melnikov, Alexander"https://zbmath.org/authors/?q=ai:melnikov.alexander-v.1|melnikov.alexander-g|melnikov.alexander-vA structure is called punctual if its basic set is the set of natural numbers or its initial segment and all its operations and relations are primitive recursive.
The authors prove that in each of the following classes, every punctual structure from the class can be punctually embedded into its punctual (existential, algebraic) closure:
(1) Boolean algebras;
(2) fields;
(3) ordered fields;
(4) differential fields of characteristic zero.
Reviewer: Andrei S. Morozov (Novosibirsk)One dimensional groups definable in the \(p\)-adic numbershttps://zbmath.org/1537.030412024-07-25T18:28:20.333415Z"Acosta López, Juan Pablo"https://zbmath.org/authors/?q=ai:acosta-lopez.juan-pabloSummary: A complete list of one dimensional groups definable in the \(p\)-adic numbers is given, up to a finite index subgroup and a quotient by a finite subgroup.Model theory of derivations of the Frobenius map revisitedhttps://zbmath.org/1537.030422024-07-25T18:28:20.333415Z"Gogolok, Jakub"https://zbmath.org/authors/?q=ai:gogolok.jakubIn this paper under review, the author shows (correcting some mistakes in previous work of \textit{P. Kowalski} [J. Symb. Log. 70, No. 1, 99--110 (200); Zbl 1094.03025)]) that the theory of fields of characteristic positive equipped with a Frobenius-derivation has a model companion, which is stable but not superstable. An explicit axiomatization similar to \textit{C. Wood}'s axioms [Isr. J. Math. 25, 331--352 (1976; Zbl 0346.02030)] for the theory \(\mathrm{DCF}_{p}\) of differentially closed fields of characteristic \(p>0\) is exhibited. In the last section, the corresponding geometric axioms as in [\textit{D. Pierce} and \textit{A. Pillay}, J. Algebra 204, No. 1, 108--115 (1998; Zbl 0922.12006)] are discussed.
A Frobenius-derivation of a field \(K\) of characteristic \(p>0\) is an additive map \(\delta\) satisfying the following identity
\[
\delta(x\cdot y)= x^p\delta(y)+y^p\delta(x).
\]
The model companion of the theory \(\mathrm{Fr}\)-\(\mathrm{DF}_p\) of fields of characteristic \(p\) equipped with a Frobenius-derivation exists and can be axiomatised by imposing the the underlying differential field is \emph{strict} (that is, the field of constants equals the \(p\)-powers) and Blum's axiom (see [\textit{L. Blum} Contrib. to Algebra, Collect. Pap. dedic. E. Kolchin, 37--61 (1977; Zbl 0368.12013)] for differential fields. In Theorem 2.4, it is shown that the theory \(\mathrm{Fr}\)-\(\mathrm{DF}_p\) admits quantifier elimination after adding a function which picks up the unique \(p\)-root for an element of derivation \(0\).
Reviewer: Amador Martin-Pizarro (Freiburg)Unitary representations of locally compact groups as metric structureshttps://zbmath.org/1537.030432024-07-25T18:28:20.333415Z"Yaacov, Itaï Ben"https://zbmath.org/authors/?q=ai:ben-yaacov.itai"Goldbring, Isaac"https://zbmath.org/authors/?q=ai:goldbring.isaacFor a locally compact group \(G\), the authors show that the class of continuous unitary representations of \(G\) is elementary (that is: axiomatizable) in the sense of continuous logic. They also relate the notion of ultraproduct in the sense of (continuous) logic with other notions of ultraproduct of representations appearing in the literature. The authors also obtain an interesting result (Theorem 4.2) about a model-theoretic characterization of Kazhdan's property (T) in a locally compact group \(G\) in terms of definability of certain sets of fixed points in the associated structure.
Reviewer: Piotr Kowalski (Wrocław)Model theory and proof theory of the global reflection principlehttps://zbmath.org/1537.030442024-07-25T18:28:20.333415Z"Łełyk, Mateusz Zbigniew"https://zbmath.org/authors/?q=ai:lelyk.mateusz-zbigniewLet \({\mathcal L}(T)\) be the language \(\{+,\times, T\}\), where \(T\) is a unary predicate symbol. \(CT^-\) is a theory in \({\mathcal L}(T)\) consisting of Peano axioms (PA) for \(+\) and \(\times\) and Tarski's compositional axioms for truth. \(CT_0\) is \(CT^-\) plus the induction schema for bounded formulas of \({\mathcal L}(T)\). By a theorem of \textit{H. Kotlarski} et al. [Can. Math. Bull. 24, 283--293 (1981; Zbl 0471.03054)], \(CT^-\) is conservative over \(PA\), and it was first observed by \textit{H. Kotlarski} [Z. Math. Logik Grundlagen Math. 32, 531--544 (1986; Zbl 0622.03025)] that \(CT_0\) is not. Many years after the publication, a serious gap in Kotlarski's proof [loc. cit.] was discovered and further research followed. This paper begins with an extensive introduction and the history of the subject. Then in Section 3, the author shows that \(CT_0\) proves the global reflection principle GRP(PA). The proof begins with an outline of Kotlarski's proof and a clear explanation of the subtle problem with it. Then, an elaborate inductive argument is given to fill the gap. (A caveat for the potential reader: the framework for the argument is \(CT^+_0\), which is \(CT_0\) in the language with symbols for all primitive recursive functions and their definitions as axioms, but this theory is formally introduced only in Part 3.3.) As a corollary, several results are combined to show that various extensions of \(CT^-\) by formalized variants of the reflexion principle are all equivalent to \(CT_0\).
Section 4 is devoted to a solution of a problem posed by \textit{L. D. Beklemishev} and \textit{F. N. Pakhomov} [Ann. Pure Appl. Logic 173, No. 5, Article ID 103093, 41 p. (2022; Zbl 07501986)]. The author shows that \(CT_0\) proves \(\Sigma_1\)-reflection over a weak truth extension of Elementary Arithmetic.
In Section 5, a model-theoretic construction is given to reprove a theorem of Kotlarski [loc. cit.] that says that \(CT_0\) is conservative over \(\omega\)-iterated reflection principle for PA. For the proof, the author invents a technique of prolonging partial inductive satisfaction classes to end extensions that is interesting on its own.
Reviewer: Roman Kossak (New York)Decomposition into special submanifoldshttps://zbmath.org/1537.030452024-07-25T18:28:20.333415Z"Fujita, Masato"https://zbmath.org/authors/?q=ai:fujita.masato|fujita.masato.1The present paper concerns studying definably complete locally o-minimal expansions of ordered groups. The author studies properties of such expansions. He introduce the notions of a special submanifold, a quasi-special submanifold and a special manifold with a tubular neighborhood and shows differences between them. The main result of the paper is a decomposition of any definable set into finitely many special manifolds with tubular neighborhoods.
Reviewer: Beibut Kulpeshov (Almaty)Pregeometry over locally o-minimal structures and dimensionhttps://zbmath.org/1537.030462024-07-25T18:28:20.333415Z"Fujita, Masato"https://zbmath.org/authors/?q=ai:fujita.masato.1|fujita.masatoThe author introduces a new closure operator, the discrete closure, leading to a pregeometry in locally o-minimal, definably complete structures. He proves that the rank function (called discr-dimension in the paper) for this pregeometry on the definable sets is equal to the geometric dimension (defined by coordinate projections) for such structures. Moreover, as the structures are first-order topological in the sense of [\textit{A. Pillay}, J. Symb. Log. 52, No. 3, 763--778 (1987; Zbl 0628.03022)] and definable sets are (topologically) constructible, the available notion of dimension rank (using closed, nowhere dense, definable subsets) is also equal to the geometric dimension. This is an extension of theorems by \textit{L. Mathews} [Proc. Lond. Math. Soc. III. Ser. 70, No. 1, 1--32 (1995; Zbl 0829.03019)] that were available for structures having cell decomposition such as o-minimal structures.
Reviewer: Artur Piękosz (Kraków)A family of dp-minimal expansions of \((\mathbb{Z}; +)\)https://zbmath.org/1537.030472024-07-25T18:28:20.333415Z"Tran, Chieu-Minh"https://zbmath.org/authors/?q=ai:tran.chieu-minh"Walsberg, Erik"https://zbmath.org/authors/?q=ai:walsberg.erikThe present paper concerns studying expansions of the additive group of integers by a ternary relation of an additive cyclic order. The authors show that such expansions are dp-minimal. They also prove that one can produce continuum of dp-minimal expansions of the additive group of integers up to definable equivalence. Thus, the authors obtain a result, which refutes a supposition that there are not too many dp-minimal expansions of this group.
Reviewer: Beibut Kulpeshov (Almaty)Metric groups, unitary representations and continuous logichttps://zbmath.org/1537.030482024-07-25T18:28:20.333415Z"Ivanov, Aleksander"https://zbmath.org/authors/?q=ai:ivanov.aleksandr-aleksandrovich|ivanov.aleksandr-pavlovich|ivanov.aleksandr-aleksandrovich.1|ivanov.aleksander|ivanov.aleksandr-gennadevich|ivanov.aleksandr-olegovich|ivanov.aleksandr-s|ivanov.aleksandr-yurevich|ivanov.aleksandr-valentinovich|ivanov.aleksandr-vladimirovich.1Summary: We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find \(L_{\omega_1 \omega}\)-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.The Diophantine problem in Chevalley groupshttps://zbmath.org/1537.030512024-07-25T18:28:20.333415Z"Bunina, Elena"https://zbmath.org/authors/?q=ai:bunina.elena-igorevna"Myasnikov, Alexei"https://zbmath.org/authors/?q=ai:myasnikov.alexei-g"Plotkin, Eugene"https://zbmath.org/authors/?q=ai:plotkin.eugene|plotkin.eugene-i|plotkin.eugene-bThe Diophantine problem \(\mathcal{D}(\mathcal{A})\) in an algebraic structure \(\mathcal{A}\), asks whether there exists an algorithm that, given a finite system \(S\) of equations in finitely many variables and coefficients in \(\mathcal{A}\), determines if \(S\) has a solution in \(\mathcal{A}\) or not.
The Diophantine problem in a structure \(\mathcal{A}\) reduces to the Diophantine problem in a structure \(\mathcal{B}\), symbolically \(\mathcal{D}(\mathcal{A}) \leq \mathcal{D}(\mathcal{B})\), if there is an algorithm that for a given finite system of equations \(S\) with coefficients in \(\mathcal{A}\) constructs a system of equations \(S^{\ast}\) with coefficients in \(\mathcal{B}\) such that \(S\) has a solution in \(\mathcal{A}\) if and only if \(S^{\ast}\) has a solution in \(\mathcal{B}\). In particular, if \(R\) is a commutative ring and \(\mathcal{D}(\mathbb{Z})\leq \mathcal{D}(R)\), then \(\mathcal{D}(R)\) is undecidable. If the reducing algorithm is polynomial-time then the reduction is termed polynomial-time (or Karp reduction).
In the paper under review, the authors study the Diophantine problem in Chevalley groups \(G_{\pi}(\Phi,R)\). They show that the Diophantine problems in \(G_{\pi}(\Phi,R)\) and \(R\) are polynomial time equivalent. They also establish a variant of a double centralizer theorem for elementary unipotents \(x_{\alpha}(1)\). This theorem is valid for arbitrary commutative rings with \(1\). The result is principal to show that any one-parametric subgroup \(X_{\alpha}\), \(\alpha \in \Phi\), is Diophantine in \(G_{\pi}(\Phi,R)\) and then to prove the main result.
Reviewer: Enrico Jabara (Venezia)Computable presentability of countable linear ordershttps://zbmath.org/1537.030522024-07-25T18:28:20.333415Z"Frolov, A. N."https://zbmath.org/authors/?q=ai:frolov.andrei-nikolaevichSummary: The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In [Ann. Pure Appl. Logic 52, No. 1--2, 39--64 (1991; Zbl 0734.03026)], \textit{C. G. Jockusch jun.} and \textit{R. I. Soare} constructed a low linear order without computable presentations. Earlier, in [Z. Math. Logik Grundlagen Math. 35, No. 3, 237--246 (1989; Zbl 0654.03032)], \textit{R. G. Downey} and \textit{M. F. Moses} showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations the existence of a computable presentation is sufficient. This question (namely, research program) was stated by \textit{R. G. Downey} in [Stud. Logic Found. Math. 139, 823--976 (1998; Zbl 0941.03045)]: Describe the order property \(P\) such that, for any low linear order \(L\), \(P(L)\) implies the existence of a computable presentation of \(L\). In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship.Bi-interpretation in weak set theorieshttps://zbmath.org/1537.030712024-07-25T18:28:20.333415Z"Roque Freire, Alfredo"https://zbmath.org/authors/?q=ai:freire.alfredo-roque"Hamkins, Joel David"https://zbmath.org/authors/?q=ai:hamkins.joel-davidSummary: In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory \(\text{ZFC}^-\) without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of \(\text{ZFC}^-\) that are bi-interpretable, but not isomorphic -- even \(\langle H_{\omega_1},\in\rangle\) and \(\langle H_{\omega_2},\in \rangle\) can be bi-interpretable -- and there are distinct bi-interpretable theories extending \(\text{ZFC}^-\) . Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.Compositional truth with propositional tautologies and quantifier-free correctnesshttps://zbmath.org/1537.030912024-07-25T18:28:20.333415Z"Wcisło, Bartosz"https://zbmath.org/authors/?q=ai:wcislo.bartoszThis is another in a series of papers by various authors on what has been dubbed by Ali Enayat the Tarski boundary problem. By the well-known theorem of \textit{H. Kotlarski} et al. [Can. Math. Bull. 24, 283--293 (1981; Zbl 0471.03054)], the theory \(\mathsf{CT}^-\), that extends Peano arithmetic (\(\mathsf{PA}\)) by compositional axioms for a truth predicate, is conservative over \(\mathsf{PA}\). The theory \(\mathsf{CT}\) obtained by adding to \(\mathsf{CT}^-\) the induction schema for the full language with the truth predicate symbol \(T\) is not a not a conservative extension of \(\mathsf{PA}\), and so is \(\mathsf{CT}_0\), which is \(\mathsf{CT}\) with the induction axioms for formulas with \(T\) is restricted to bounded formulas.
Many seemingly innocuous extensions of \(\mathsf{CT}^-\) are pushed into the nonconservative side and often those extensions turn out to be equivalent to \(\mathsf{CT}_0\). By a result of \textit{A. Enayat} and \textit{F. Pakhomov} [Arch. Math. Logic 58, No. 5--6, 753--766 (2019; Zbl 1477.03250)], prominent among them is obtained by adding to \(\mathsf{CT}^-\) the principle of disjunctive correctness \(\mathsf{DC}\).
The main result of this paper adds to the list \(\mathsf{CT}^-\) extended by the propositional soundness principle \(\mathsf{PS}\) that states that any propositional tautology is true (according to the truth predicate). The proof is in two parts. It is first shown that \(\mathsf{CT}^-+\mathsf{PS}\) extended by the quantifier-free correctness principle \(\mathsf{QFC}\) is equivalent to \(\mathsf{CT}_0\) and to prove this the author applies his own technique of disjunctions with stopping conditions. Then, \(\mathsf{QFC}\) is eliminated by showing that \(\mathsf{CT}^-+\mathsf{QFC}\) is conservative over \(\mathsf{PA}\). This second part uses a variant of the Enayat-Visser construction of satisfaction classes introduced in [\textit{A. Enayat} and \textit{A. Visser}, Log. Epistemol. Unity Sci. 36, 321--335 (2015; \url{doi:10.1007/978-94-017-9673-6_16})].
Despite relaying on the results by Enayat and Pakhomov [loc. cit.], and Enayat and Visser [loc. cit.], the paper is self-contained and full proofs of the variants of these results are given in two appendices. The paper is very well written and it can serve as a good introduction to the Tarski boundary problem.
Reviewer: Roman Kossak (New York)Finiteness for self-dual classes in integral variations of Hodge structurehttps://zbmath.org/1537.140152024-07-25T18:28:20.333415Z"Bakker, Benjamin"https://zbmath.org/authors/?q=ai:bakker.benjamin"Grimm, Thomas W."https://zbmath.org/authors/?q=ai:grimm.thomas-w"Schnell, Christian"https://zbmath.org/authors/?q=ai:schnell.christian"Tsimerman, Jacob"https://zbmath.org/authors/?q=ai:tsimerman.jacobSummary: We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure \(\mathbb{R}_{\text{an,exp}}\).Canonical stratification of definable Lie groupoidshttps://zbmath.org/1537.140802024-07-25T18:28:20.333415Z"Tanabe, Masato"https://zbmath.org/authors/?q=ai:tanabe.masatoSummary: Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, \(\mathrm{o}\)-minimal over \(\mathbb{R}\), or more generally, Shiota's \(\mathfrak{X}\)-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J. N. Mather and V. A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.Generic types and generic elements in divisible rigid groupshttps://zbmath.org/1537.200832024-07-25T18:28:20.333415Z"Myasnikov, A. G."https://zbmath.org/authors/?q=ai:myasnikov.alexei-g"Romanovskii, N. S."https://zbmath.org/authors/?q=ai:romanovskii.n-sAuthors' abstract: A group \(G\) is said to be \(m\)-rigid if it contains a normal series of the form \(G = G_1 > G_2 > \dots > G_m > G_{m+1} = 1\), whose quotients \(G_i/G_{i+1}\) are Abelian and, treated as (right) \(\mathbb{Z}[G/G_i]\)-modules, are torsion-free. A rigid group \(G\) is said to be divisible if elements of the quotient \(\rho_i(G)/\rho_{i+1}(G)\) are divisible by nonzero elements of the ring \(\mathbb{Z}[G/\rho_i(G)]\). Previously, it was proved that the theory of divisible \(m\)-rigid groups is complete and \(\omega\)-stable. In the present paper, we give an algebraic description of elements and types that are generic over a divisible \(m\)-rigid group \(G\).
Reviewer: Alexander Ivanovich Budkin (Barnaul)Convergent sequences in groups and idealshttps://zbmath.org/1537.220012024-07-25T18:28:20.333415Z"Hrušák, Michael"https://zbmath.org/authors/?q=ai:hrusak.michael"Shibakov, Alexander"https://zbmath.org/authors/?q=ai:shibakov.alexander-yIf \(X\) is a topological space and \(S\subseteq X\) is an infinite countable subset then the same authors say that \(S\) converges to some \(x\in X\) if \(S\setminus U\) is finite for any neighborhood \(U\) of \(x\) in~\(X\). In [\textit{M. Hrušák} and \textit{A. Shibakov}, Forum Math. Sigma 10, Paper No. e29, 18 p. (2022; Zbl 07537758)], the authors have introduced the so-called invariant ideal axiom. In the present paper, the authors review some recent results concerning the convergence in topological groups with a special emphasis on the consequences of the invariant ideal axiom. Moreover, they present some open problems.
Reviewer: Alexander Isaakovich Shtern (Moskva)Potential isomorphisms of generalized approach spaceshttps://zbmath.org/1537.540032024-07-25T18:28:20.333415Z"Ackerman, Nathanael"https://zbmath.org/authors/?q=ai:ackerman.nathanael-leedom"Karker, Mary Leah"https://zbmath.org/authors/?q=ai:karker.mary-leahIn this paper the authors study potential isomorphisms of approach spaces valued in a quantale. Approach spaces, i.e. $[0,\infty]$-valued approach spaces were discovered by Lowen as a common generalization of metric and topological spaces. But unfortunately the definition of a generalized approach space is not first order, meaning that it places structure not just on the underlying set but also on the collection of subsets of an underlying set. In improving the situation the authors show that any generalized approach space has a relativization as a generalized approach space in each larger model of set theory. Furthermore, the authors give a necessary and sufficient condition for the existence of a potential isomorphism between generalized approach spaces which is absolute between models of set theory. In addition an abstract notion of sentence for generalized approach spaces is considered as a pair $(A, \alpha)$, where $A$ is a generalized approach basis, $\alpha$ an ordinal and where a generalized approach basis $B$ satisfies $(A, \alpha)$ if and only if there is a back and forth system between $B$ and $A$ of length $\alpha$. Then the paper concludes with proving a downwards Löwenheim-Skolem type theorem as well as a Tarski-Vaught type theorem for collections of these sentences.
Reviewer: Dieter Leseberg (Berlin)Three parameter metrics in the presence of a scalar field in four and higher dimensionshttps://zbmath.org/1537.810102024-07-25T18:28:20.333415Z"Azizallahi, Alireza"https://zbmath.org/authors/?q=ai:azizallahi.alireza"Mirza, Behrouz"https://zbmath.org/authors/?q=ai:mirza.behrouz"Hajibarat, Arash"https://zbmath.org/authors/?q=ai:hajibarat.arash"Anjomshoa, Homayon"https://zbmath.org/authors/?q=ai:anjomshoa.homayonSummary: We investigate a class of three parameter metrics that contain both the \(\gamma\)-metric and Janis-Newman-Winicour (JNW) metric at special values of the parameters. To see the effect of the scalar field we derive some properties of this class of metrics such as curvature invariants, the effective potential, and epicyclic frequencies. We also introduce the five and higher dimensional forms of the class of metrics in the presence of a scalar field.Singularities of 1/2 Calabi-Yau 4-folds and classification scheme for gauge groups in four-dimensional F-theoryhttps://zbmath.org/1537.810132024-07-25T18:28:20.333415Z"Kimura, Yusuke"https://zbmath.org/authors/?q=ai:kimura.yusukeSummary: In a previous study, we constructed a family of elliptic Calabi-Yau 4 folds possessing a geometric structure that allowed them to be split into a pair of rational elliptic 4 folds. In the present study, we introduce a method of classifying the singularity types of this class of elliptic Calabi-Yau 4 folds. In brief, we propose a method to classify the non-Abelian gauge groups formed in four-dimensional (4D) \(N =1\) F-theory for this class of elliptic Calabi-Yau 4 folds.
To demonstrate our method, we explicitly construct several elliptic Calabi-Yau 4 folds belonging to this class and study the 4D F-theory thereupon. These constructions include a 4D model with two U(1) factors.Insights into quantum contextuality and Bell nonclassicality: a study on random pure two-qubit systemshttps://zbmath.org/1537.810172024-07-25T18:28:20.333415Z"Scala, Giovanni"https://zbmath.org/authors/?q=ai:scala.giovanni"Mandarino, Antonio"https://zbmath.org/authors/?q=ai:mandarino.antonioSummary: We explore the relationship between Kochen-Specker quantum contextuality and Bell-nonclassicality for ensembles of two-qubit pure states. We present a comparative analysis showing that the violation of a noncontextuality inequality on a given quantum state reverberates on the Bell-nonclassicality of the considered state. In particular, we use suitable inequalities that are experimentally testable to detect quantum contextuality and nonlocality for systems in a Hilbert space of dimension \(\boldsymbol{d=4}\). While contextuality can be assessed on different degrees of freedom of the same particle, the violation of local realism requires parties spatially separated.Fermi isospectrality for discrete periodic Schrödinger operatorshttps://zbmath.org/1537.810862024-07-25T18:28:20.333415Z"Liu, Wencai"https://zbmath.org/authors/?q=ai:liu.wencaiSummary: Let \(\Gamma =q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus \ldots \oplus q_d\mathbb{Z}\), where \(q_l\in \mathbb{Z}_+\), \(l=1,2,\ldots ,d\), are pairwise coprime. Let \(\Delta +V\) be the discrete Schrödinger operator, where \(\Delta\) is the discrete Laplacian on \(\mathbb{Z}^d\) and the potential \(V:\mathbb{Z}^d\rightarrow \mathbb{C}\) is \(\Gamma\)-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension \(d\ge 3\):
\begin{itemize}
\item[(1)] If at some energy level, Fermi varieties of two real-valued \(\Gamma\)-periodic potentials \(V\) and \(Y\) are the same (this feature is referred to as \textit{Fermi isospectrality} of \(V\) and \(Y)\), and \(Y\) is a separable function, then \(V\) is separable;
\item[(2)] If two complex-valued \(\Gamma\)-periodic potentials \(V\) and \(Y\) are Fermi isospectral and both \(V=\bigoplus_{j=1}^rV_j\) and \(Y=\bigoplus_{j=1}^r Y_j\) are separable functions, then, up to a constant, lower dimensional decompositions \(V_j\) and \(Y_j\) are Floquet isospectral, \(j=1,2,\ldots ,r\);
\item[(3)] If a real-valued \(\Gamma\)-potential \(V\) and the zero potential are Fermi isospectral, then \(V\) is zero.
\end{itemize}
In particular, all conclusions in (1), (2) and (3) hold if we replace the assumption ``Fermi isospectrality'' with a stronger assumption ``Floquet isospectrality''.
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