Recent zbMATH articles in MSC 03Chttps://zbmath.org/atom/cc/03C2022-07-25T18:03:43.254055ZUnknown authorWerkzeugHomogeneous structures: model theory meets universal algebra. Abstracts from the workshop held January 3--9, 2021 (online meeting)https://zbmath.org/1487.000322022-07-25T18:03:43.254055ZSummary: The workshop ``Homogeneous Structures: Model Theory meets Universal Algebra'' was centred around transferring recently obtained advances in universal algebra from the finite to the infinite. As it turns out, the notion of homogeneity together with other model-theoretic concepts like \(\omega\)-categoricity and the Ramsey property play an indispensable role in this endeavour.And what if Mara's models don't have elements?https://zbmath.org/1487.030502022-07-25T18:03:43.254055Z"Hernández-Manfredini, Enrique"https://zbmath.org/authors/?q=ai:hernandez-manfredini.enrique"Martins, Manuel A."https://zbmath.org/authors/?q=ai:martins.manuel-a
For the entire collection see [Zbl 1446.03002].From pragmatic truths to emotional truthshttps://zbmath.org/1487.030512022-07-25T18:03:43.254055Z"Schumann, Andrew"https://zbmath.org/authors/?q=ai:schumann.andrewSummary: In this paper, I propose an extension of partial models introduced first by \textit{I. Mikenberg} et al. [J. Symb. Log. 51, 201--221 (1986; Zbl 0606.03009)]. The partial models were made to explicate the notion of pragmatic truths. In my extension of partial models we can define partial relations for explicating the notion of emotional truths.
For the entire collection see [Zbl 1455.03001].Eliminating field quantifiers in strongly dependent Henselian fieldshttps://zbmath.org/1487.030522022-07-25T18:03:43.254055Z"Halevi, Yatir"https://zbmath.org/authors/?q=ai:halevi.yatir"Hasson, Assaf"https://zbmath.org/authors/?q=ai:hasson.assafRecall that, following Shelah, a first-order structure \(\mathfrak A\) is called strongly dependent if there are \textbf{no} model \(\mathfrak M\equiv \mathfrak A\), formulas \(\phi_i(x,y)\) for \(i\in\omega\), and elements \(a^j_i\in \mathfrak M \) so that for each function \(\eta:\omega \to \omega\) the type \[\{\phi_i(x,a^j_i) : i, j\in \omega, \eta(i)=j\} \cup \{\neg\phi_i(x,a^j_i) : i, j\in \omega, \eta(i)\ne j\}\] is consistent. Such structures are now considered as ``tame'' settings.
From the Introduction: ``Shelah's conjecture [\textit{S. Shelah}, Isr. J. Math. 173, 1--60 (2009; Zbl 1195.03040)], usually interpreted as stating that strongly dependent fields which are neither real closed nor algebraically closed are Henselian, is our main motivation for carrying out the present research''.
Summary: ``We prove the elimination of field quantifiers for strongly dependent Henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if \((K, v)\) is strongly dependent, then so is its Henselization.''
Enough properties of strongly dependent Henselian valued fields are known so that the authors can use them directly. The authors rely mostly on \textit{W. Johnson}'s thesis [Fun with fields. Berkeley, CA: University of California (PhD Thesis) (2016)].
Reviewer: Luc Bélair (Montréal)Axiomatizability of the class of subdirectly irreducible acts over an abelian grouphttps://zbmath.org/1487.030532022-07-25T18:03:43.254055Z"Stepanova, A. A."https://zbmath.org/authors/?q=ai:stepanova.alena-andreevna"Ptakhov, D. O."https://zbmath.org/authors/?q=ai:ptakhov.d-oThe main result describes abelian groups over which the class of all subdirectly irreducible acts is axiomatizable as follows:
The class \(SIr(G)\) of all subdirectly irreducible acts over an abelian group \(G\) is axiomatizable if and only if there exist \(k\in\omega\) and a finite set \(T\subseteq G\) such that any subdirectly irreducible group \(G/H\), where \(H\) is a subgroup of \(G\), is a cyclic group of order \(p^m \le k\) (here \(m\in\omega\), \(p\) is a prime) generated by an element \(gH\in \{tH \mid t \in T \}\).
Moreover, some properties of subdirectly irreducible acts over abelian groups are studied. It is proved that all connected acts over an abelian group are subdirectly irreducible iff the group is totally ordered. Note that a group \(G\) is said to be linearly (totally) ordered if the set \(\{H \mid H \le G\}\) is linearly (totally) ordered under \(\subseteq\).
Reviewer: Ulrich Knauer (Oldenburg)Logical structures from a model-theoretical viewpointhttps://zbmath.org/1487.030542022-07-25T18:03:43.254055Z"Beziau, Jean-Yves"https://zbmath.org/authors/?q=ai:beziau.jean-yvesSummary: We first explain what it means to consider logics as structures. In a second part we discuss the relation between structures and axioms, explaining in particular what axiomatization from a model-theoretical perspective is. We then go on by discussing the place of logical structures among other mathematical structures and by giving an outlook on the varied universe of logical structures. After that we deal with axioms for logical structures, in a first part in an abstract setting, in a second part dealing with negation. We end by saying a few words about Edelcio.
For the entire collection see [Zbl 1455.03001].Suppes predicate for classes of structures and the notion of transportabilityhttps://zbmath.org/1487.030552022-07-25T18:03:43.254055Z"da Costa, Newton C. A."https://zbmath.org/authors/?q=ai:da-costa.newton-carneiro-affonso"Krause, Decio"https://zbmath.org/authors/?q=ai:krause.decioSummary: Patrick Suppes' maxim ``to axiomatize a theory is to define a set-theoretical predicate'' is usually taking as entailing that the formula that defines the predicate needs to be transportable in the sense of Bourbaki. We argue that this holds for theories, where we need to cope with all structures (the models) satisfying the predicate. For instance, in axiomatizing the theory of groups, we need to grasp all groups. But we may be interested in catching not all structures of a species, but just some of them. In this case, the formula that defines the predicate doesn't need to be transportable. The study of this question has lead us to a careful consideration of Bourbaki's definition of transportability, usually not found in the literature. In this paper we discuss this topic with examples, recall the notion of transportable formulas and show that we can have significant set-theoretical predicates for classes of structures defined by non transportable formulas as well.
For the entire collection see [Zbl 1455.03001].Classes of barren extensionshttps://zbmath.org/1487.030562022-07-25T18:03:43.254055Z"Dobrinen, Natasha"https://zbmath.org/authors/?q=ai:dobrinen.natasha-l"Hathaway, Dan"https://zbmath.org/authors/?q=ai:hathaway.danThe infinite partition relation \(\omega \rightarrow (\omega)^{\omega}\) means that
for each coloring \(c : [\omega]^{\omega} \rightarrow 2\) there is an infinite subset
\(x \subseteq \omega\) such that the restriction of \(c\) to \([x]^{\omega}\) is constant.
\textit{J. M. Henle} et al. [Lect. Notes Math. 1130, 195--207 (1985; Zbl 0594.03030)] proved the following: If \(M\) is a transitive model of
\(\text{ZF} + \omega \rightarrow (\omega)^{\omega}\) and \(N\) is its extension via
\(([\omega]^{\omega},\subseteq^*)\), then \(M\) and \(N\) have the same sets of
ordinals and every well-ordered sequence in \(N\) of elements of \(M\) lies in \(M\).
Forcing with \(([\omega]^{\omega},\subseteq^*)\) adds a Ramsey ultrafilter.
They called forcing relations which do not add any new subsets of ordinals over
the ground model, \textit{barren} extension.
Henle et al. [loc. cit.] formulated two additional properties, called LU and EP and
showed that under these additional assumptions, forcing with
\(([\omega]^{\omega},\subseteq^*)\) preserves all strong partition cardinals.
Here, the authors investigate the following question: Are there forcing relations
which add non-Ramsey ultrafilters for which the consequences of the above theorems of
Henle et al. [loc. cit.] still hold? They show that several classes of \(\sigma\)-closed forcings
generating non-Ramsey ultrafilters have these properties. Such ultrafilters include
Milliken-Taylor ultrafilters, a class of rapid p-points of \textit{C. Laflamme} [Ann. Pure Appl. Logic 42, No. 2, 125--163 (1989; Zbl 0681.03035)], \(k\)-arrow p-points
of \textit{J. E. Baumgartner} and \textit{A. D. Taylor} [Trans. Am. Math. Soc. 241, 283--309 (1978; Zbl 0386.03024)] and extensions of ultrafilters constructed by \textit{N. Dobrinen} et al. [Arch. Math. Logic 56, No. 7--8, 733--782 (2017; Zbl 1417.03245)]. They show that forcing with the Boolean algebras
\(\mathcal{P}(\omega^{\alpha})/\mathrm{Fin}^{\otimes\alpha}\) for \(2 \leq \alpha <\omega_1\)
generate non-p-points and produce barren extensions.
Reviewer: Martin Weese (Potsdam)An extremal problem motivated by triangle-free strongly regular graphshttps://zbmath.org/1487.051412022-07-25T18:03:43.254055Z"Razborov, Alexander"https://zbmath.org/authors/?q=ai:razborov.alexander-aSummary: We introduce the following combinatorial problem. Let \(G\) be a triangle-free regular graph with edge density \(\rho \). (In this paper all densities are normalized by \(n, \frac{n^2}{2}\) etc. rather than by \(n - 1, \binom{n}{2}, \ldots)\) What is the minimum value \(a(\rho)\) for which there always exist two non-adjacent vertices such that the density of their common neighbourhood is \(\leq a(\rho)\)? We prove a variety of upper bounds on the function \(a(\rho)\) that are tight for the values \(\rho = 2 / 5, 5 / 16, 3 / 10, 11 / 50\), with \(C_5\), Clebsch, Petersen and Higman-Sims being respective extremal configurations. Our proofs are entirely combinatorial and are largely based on counting densities in the style of flag algebras. For small values of \(\rho \), our bound attaches a combinatorial meaning to so-called Krein conditions that might be interesting in its own right. We also prove that for any \(\epsilon > 0\) there are only finitely many values of \(\rho\) with \(a(\rho) \geq \epsilon\) but this finiteness result is somewhat purely existential (the bound is double exponential in \(1 / \epsilon )\).Induced and higher-dimensional stable independencehttps://zbmath.org/1487.180052022-07-25T18:03:43.254055Z"Lieberman, Michael"https://zbmath.org/authors/?q=ai:lieberman.michael-j"Rosický, Jiří"https://zbmath.org/authors/?q=ai:rosicky.jiri"Vasey, Sebastien"https://zbmath.org/authors/?q=ai:vasey.sebastienSummary: We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [\textit{M. Mazari-Armida}, J. Algebra 567, 196--209 (2021; Zbl 07282595)]. We prove, too, that under the hypotheses of [\textit{M. Lieberman} et al., ``Cellular categories and stable independence'', submitted for publication], a stable independence notion immediately yields higher-dimensional independence as in [\textit{S. Shelah} and \textit{S. Vasey}, ``Categoricity and multidimensional diagrams'', Preprint, \url{arXiv:1805.06291}].The Diophantine problem in the classical matrix groupshttps://zbmath.org/1487.200112022-07-25T18:03:43.254055Z"Myasnikov, Aleksei G."https://zbmath.org/authors/?q=ai:myasnikov.alexei-g"Sohrabi, Mahmood"https://zbmath.org/authors/?q=ai:sohrabi.mahmoodLet \(R\) be a countable (computable) associative ring with identity and let \(G\) be one of the groups \(\mathrm{GL}_n(R)\), \(T_n(R)\), \(UT_n(R)\), \(n\geq3\). Here, \(T_n(R)\) is the upper triangular subgroup of \(\mathrm{GL}_n(R)\) and \(UT_n(R)\) is the unipotent subgroup of \(T_n(R)\). By a Diophantine problem in \(G\) we mean the problem to decide if a finite system of group equations has a solution in \(G\). Here, a group equation is of the form \(w(x_1, \dots , x_n, g_1, \dots , g_m) = 1\), where \(w\) is a group word in variables \(x_1, \dots , x_n\) and constants \(g_1, \dots , g_m \in G\). Similarly one has Diophantine problems in \(R\) with equations that are polynomials with constants in \(R\). The main result of the paper tells that a Diophantine problem in \(G\) is polynomial-time equivalent (more precisely, Karp equivalent) to a Diophantine problem in \(R\). Similar results hold for \(\mathrm{PGL}_n(R)\), and, if \(R\) is commutative, \(\mathrm{SL}_n(R)\), \(\mathrm{PSL}_n(R)\). One also considers subgroups of \(G\) that contain all elementary matrices in \(G\). In fact a key point is that the ring structure on \(R\) may be reconstructed from the group multiplication of elementary matrices. And a standard one parameter elementary subgroup is Diophantine in \(G\), i.e. definable by group equations.
The text explains the context and provides ample examples, both of decidable and undecidable problems.
Reviewer: Wilberd van der Kallen (Utrecht)Factorial relative commutants and the generalized Jung property for \(\mathrm{II}_1\) factorshttps://zbmath.org/1487.460682022-07-25T18:03:43.254055Z"Atkinson, Scott"https://zbmath.org/authors/?q=ai:atkinson.scott-e"Goldbring, Isaac"https://zbmath.org/authors/?q=ai:goldbring.isaac"Kunnawalkam Elayavalli, Srivatsav"https://zbmath.org/authors/?q=ai:kunnawalkam-elayavalli.srivatsavThis paper develops and uses techniques from continuous model theory to approach questions about embeddings of II\(_1\)-factors (certain von Neumann algebras with trivial center) in their ultrapowers.
Unarguably, the most important II\(_1\) factor is the hyperfinite II\(_1\) factor \(\mathcal R\), the only separable object among amenable II\(_1\) factors. A key property of the \(\mathcal R\) is that any two embeddings of \(\mathcal R\) into its ultrapower \(\mathcal R^\mathcal U\) are unitarily equivalent. Indeed, a striking result of \textit{K. Jung} [Math. Ann. 338, No. 1, 241--248 (2007; Zbl 1121.46052)] gives that \(\mathcal R\) is the only II\(_1\)-factor with this property, that is, if a II\(_1\)-factor \(N\) has the property that all embeddings of \(N\) into \(\mathcal R^\mathcal U\) are unitarily equivalent, then \(N\cong \mathcal R\).
The main focus of the present paper is a generalisation of the equivalence relation considered in Jung's theorem: Instead of focusing on automorphisms modulo unitarily equivalence, one focuses on the equivalence relations of `being conjugated by an automorphisms of the range structure'.
To formalise this, the authors introduce the definitions of Jung pair and generalised Jung pair. In their terminology, a pair of II\(_1\)-factors \((N,M)\) is a Jung pair if all embeddings of \(N\) into \(M^\mathcal U\) are unitarily equivalent, while the generalised notion asks for equality modulo an automorphism of \(M^\mathcal U\). In this setting, Jung's theorem asserts that if \((N,\mathcal R)\) is a Jung pair, then \(N\cong \mathcal R\). The authors prove the same for generalised Jung pairs, that is, that the only II\(_1\) factor \(N\) with the property that all embeddings of \(N\) into \(\mathcal R^\mathcal U\) are equivalent modulo an automorphism of \(\mathcal R^\mathcal U\) is \(\mathcal R\).
Next, attention is paid to non-\(\mathcal R^\mathcal U\)-embeddable II\(_1\)-factors. A II\(_1\)-factor which cannot be embedded in an ultrapower of \(\mathcal R\) is said to be non-\(\mathcal R^\mathcal U\)-embeddable. Such II\(_1\)-factors exist by the recent negative answer to the Connes Embedding Problem [\textit{Z.-F. Ji} et al., ``\textsf{MIP}\(^*\)=\textsf{RE}'', Preprint (2020), \url{arxiv:2001.04383}]. The existence of a non-\(\mathcal R^\mathcal U\)-embeddable II\(_1\)-factor \(M\) such that \((M,M)\) forms a generalised Jung pair is proved in \S3. This gives examples of II\(_1\)-factors which, although being non-\(\mathcal R^\mathcal U\)-embeddable, resemble one of the key property of \(\mathcal R\) itself.
Further, the authors study factorial embeddings. (An embedding of \(N\) into \(\mathcal M^{\mathcal U}\) is factorial if the relative commutant of the image of \(N\) is a factor itself.) Popa asked whether all \(\mathcal R^\mathcal U\)-embeddable factors admit a factorial embedding into \(\mathcal R^\mathcal U\). While \(\mathcal R\) is the only II\(_1\)-factor such that \emph{all} embeddings in \(\mathcal R^\mathcal U\) are factorial (by a result of \textit{N. P. Brown} [Adv. Math. 227, No. 4, 1665--1699 (2011; Zbl 1229.46041)]), there were few examples so far of II\(_1\)-factors which are \(\mathcal R^\mathcal I\)-embeddable and admit a factorial embedding into \(\mathcal R^\mathcal U\). The authors add to these, by showing that every II\(_1\)-factor which is elementary equivalent to \(\mathcal R\) has a factorial embedding into \(\mathcal R^\mathcal U\) (stronger than this, they prove that all elementary embeddings \(N\to\mathcal R^\mathcal U\) are factorial). This gives continuum many nonisomorphic II\(_1\)-factors admitting factorial embeddings into \(\mathcal R^\mathcal U\). (The study of factorial embeddings and Popa's question from a model theoretic point of view was already initiated in [\textit{I. Goldbring}, Proc. Am. Math. Soc. 148, No. 11, 5007--5012 (2020; Zbl 1456.03058)], starting from ideas of [\textit{I. Farah} et al., Fundam. Math. 233, No. 2, 173--196 (2016; Zbl 1436.03213)].)
The study of (generalised) Jung pairs is done via the study of the structure of all homomorphisms of a II\(_1\)-factor \(N\) into the ultrapower of a second II\(_1\)-factor \(M\), modulo automorphism equivalence. This set, denoted \(\mathrm{Hom}_A(N,M^\mathcal U)\), is studied from a model theoretic point of view, by associating it to a certain space of types in \S4 and studying its structure as a topometric space, following [\textit{I. Ben Yaacov}, Log. Anal. 1, No. 3--4, 235--272 (2008; Zbl 1180.03040)].
Reviewer: Alessandro Vignati (Paris)Symbolic optimization of algebraic functionshttps://zbmath.org/1487.682562022-07-25T18:03:43.254055Z"Kanno, Masaaki"https://zbmath.org/authors/?q=ai:kanno.masaaki"Yokoyama, Kazuhiro"https://zbmath.org/authors/?q=ai:yokoyama.kazuhiro"Anai, Hirokazu"https://zbmath.org/authors/?q=ai:anai.hirokazu"Hara, Shinji"https://zbmath.org/authors/?q=ai:hara.shinjiDerivative expansion in the HAL QCD method for a separable potentialhttps://zbmath.org/1487.811032022-07-25T18:03:43.254055Z"Aoki, Sinya"https://zbmath.org/authors/?q=ai:aoki.sinya"Yazaki, Koichi"https://zbmath.org/authors/?q=ai:yazaki.koichi(no abstract)