Pub Date : 2025-05-22DOI: 10.1016/j.apal.2025.103613
Sourayan Banerjee, Amit Kuber
Motivated by Krajiček and Scanlon's definition of the Grothendieck ring of a first-order structure M, we introduce the definition of K-groups for via Quillen's construction. We provide a recipe for the computation of , where is a free module over a PID R, subject to the knowledge of the abelianizations of the general linear groups . As a consequence, we provide explicit computations of when R belongs to a large class of Euclidean domains that includes fields with at least 3 elements and polynomial rings over fields with characteristic 0. We also show that the algebraic of a PID R embeds into .
{"title":"Model-theoretic K1 of free modules over PIDs","authors":"Sourayan Banerjee, Amit Kuber","doi":"10.1016/j.apal.2025.103613","DOIUrl":"10.1016/j.apal.2025.103613","url":null,"abstract":"<div><div>Motivated by Krajiček and Scanlon's definition of the Grothendieck ring <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> of a first-order structure <em>M</em>, we introduce the definition of <em>K</em>-groups <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> via Quillen's <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>S</mi></math></span> construction. We provide a recipe for the computation of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> is a free module over a PID <em>R</em>, subject to the knowledge of the abelianizations of the general linear groups <span><math><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. As a consequence, we provide explicit computations of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span> when <em>R</em> belongs to a large class of Euclidean domains that includes fields with at least 3 elements and polynomial rings over fields with characteristic 0. We also show that the algebraic <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of a PID <em>R</em> embeds into <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103613"},"PeriodicalIF":0.6,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-21DOI: 10.1016/j.apal.2025.103614
Yaroslav Alekseev , Edward A. Hirsch
The (extended) Binary Value Principle (, the equation for and Boolean variables ) has received a lot of attention recently, several lower bounds have been proved for it [1], [2], [11]. Also it has been shown [1] that the probabilistically verifiable Ideal Proof System () [8] together with polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with an algebraic version of Tseitin's extension rule () that introduces a new variable for any polynomial. Contrary to , this is a Cook–Reckhow proof system. We show that in this context still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule [6], which is in sharp contrast with the result of [2] that shows an exponential lower bound on the size of derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that probably does not help in proving exponential lower bounds for Boolean formulas: we show that an (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from must be of exponential size.
{"title":"The power of the Binary Value Principle","authors":"Yaroslav Alekseev , Edward A. Hirsch","doi":"10.1016/j.apal.2025.103614","DOIUrl":"10.1016/j.apal.2025.103614","url":null,"abstract":"<div><div>The (extended) Binary Value Principle (<span><math><mi>eBVP</mi></math></span>, the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>−</mo><mi>k</mi></math></span> for <span><math><mi>k</mi><mo>></mo><mn>0</mn></math></span> and Boolean variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>) has received a lot of attention recently, several lower bounds have been proved for it <span><span>[1]</span></span>, <span><span>[2]</span></span>, <span><span>[11]</span></span>. Also it has been shown <span><span>[1]</span></span> that the probabilistically verifiable Ideal Proof System (<span><math><mi>IPS</mi></math></span>) <span><span>[8]</span></span> together with <span><math><mi>eBVP</mi></math></span> polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with an algebraic version of Tseitin's extension rule (<span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span>) that introduces a new variable for any polynomial. Contrary to <span><math><mi>IPS</mi></math></span>, this is a Cook–Reckhow proof system. We show that in this context <span><math><mi>eBVP</mi></math></span> still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule <span><span>[6]</span></span>, which is in sharp contrast with the result of <span><span>[2]</span></span> that shows an exponential lower bound on the size of <span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span> derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that <span><math><mi>eBVP</mi></math></span> probably does not help in proving exponential lower bounds for Boolean formulas: we show that an <span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span> (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from <span><math><mi>eBVP</mi></math></span> must be of exponential size.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103614"},"PeriodicalIF":0.6,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144195892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-14DOI: 10.1016/j.apal.2025.103611
Djamel Eddine Amir, Mathieu Hoyrup
In this article, we investigate the descriptive complexity of topological invariants. Our main goal is to understand the expressive power of low complexity invariants, by investigating which spaces they can distinguish. We study the invariants in the first two levels of the Borel hierarchy. We develop techniques to establish whether two spaces can be separated by invariants in these levels. We show that they are sufficient to separate finite topological graphs. We finally identify the complexity of recognizing the line segment.
{"title":"Descriptive complexity of topological invariants","authors":"Djamel Eddine Amir, Mathieu Hoyrup","doi":"10.1016/j.apal.2025.103611","DOIUrl":"10.1016/j.apal.2025.103611","url":null,"abstract":"<div><div>In this article, we investigate the descriptive complexity of topological invariants. Our main goal is to understand the expressive power of low complexity invariants, by investigating which spaces they can distinguish. We study the invariants in the first two levels of the Borel hierarchy. We develop techniques to establish whether two spaces can be separated by invariants in these levels. We show that they are sufficient to separate finite topological graphs. We finally identify the complexity of recognizing the line segment.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103611"},"PeriodicalIF":0.6,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-12DOI: 10.1016/j.apal.2025.103612
Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov
We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the hierarchy. We focus on linear orderings. We show that at the level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the level for any the set of linear orderings with generically or coarsely computable copies is -complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.
{"title":"Generically computable linear orderings","authors":"Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov","doi":"10.1016/j.apal.2025.103612","DOIUrl":"10.1016/j.apal.2025.103612","url":null,"abstract":"<div><div>We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>β</mi></mrow></msub></math></span> hierarchy. We focus on linear orderings. We show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span> level for any <span><math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>C</mi><mi>K</mi></mrow></msubsup></math></span> the set of linear orderings with generically or coarsely computable copies is <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103612"},"PeriodicalIF":0.6,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-08DOI: 10.1016/j.apal.2025.103609
Rafał Filipów, Adam Kwela
We examine topological spaces not distinguishing ideal pointwise and ideal σ-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number ) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal σ-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal σ-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal σ-uniform convergence of sequences of real-valued continuous functions defined on them.
{"title":"Spaces not distinguishing ideal pointwise and σ-uniform convergence","authors":"Rafał Filipów, Adam Kwela","doi":"10.1016/j.apal.2025.103609","DOIUrl":"10.1016/j.apal.2025.103609","url":null,"abstract":"<div><div>We examine topological spaces not distinguishing ideal pointwise and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number <span><math><mi>b</mi></math></span>) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal <em>σ</em>-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal <em>σ</em>-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103609"},"PeriodicalIF":0.6,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-08DOI: 10.1016/j.apal.2025.103610
Tomasz Kowalski , Katarzyna Słomczyńska
We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra terms, simpler proofs of several existing results. As a by-product, we obtain an isomorphism between the free pseudocomplemented semilattice and the poset of join-irreducibles of the free p-algebra augmented by zero.
{"title":"Free p-algebras revisited: An algebraic investigation of implication-free intuitionism","authors":"Tomasz Kowalski , Katarzyna Słomczyńska","doi":"10.1016/j.apal.2025.103610","DOIUrl":"10.1016/j.apal.2025.103610","url":null,"abstract":"<div><div>We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra terms, simpler proofs of several existing results. As a by-product, we obtain an isomorphism between the free pseudocomplemented semilattice and the poset of join-irreducibles of the free p-algebra augmented by zero.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103610"},"PeriodicalIF":0.6,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-05DOI: 10.1016/j.apal.2025.103607
Arturo Martínez-Celis, Tomasz Żuchowski
Given a function , a set is free for f if is finite. For a class of functions , we define as the smallest size of a family such that for every there is a set which is free for f, and as the smallest size of a family such that for every there is such that A is not free for f. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from [20] and [2].
{"title":"On cardinal invariants related to Rosenthal families and large-scale topology","authors":"Arturo Martínez-Celis, Tomasz Żuchowski","doi":"10.1016/j.apal.2025.103607","DOIUrl":"10.1016/j.apal.2025.103607","url":null,"abstract":"<div><div>Given a function <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, a set <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is <em>free for f</em> if <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo><mo>∩</mo><mi>A</mi></math></span> is finite. For a class of functions <span><math><mi>Γ</mi><mo>⊆</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, we define <span><math><msub><mrow><mi>ros</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> such that for every <span><math><mi>f</mi><mo>∈</mo><mi>Γ</mi></math></span> there is a set <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> which is free for <em>f</em>, and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>F</mi><mo>⊆</mo><mi>Γ</mi></math></span> such that for every <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> there is <span><math><mi>f</mi><mo>∈</mo><mi>F</mi></math></span> such that <em>A</em> is not free for <em>f</em>. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from <span><span>[20]</span></span> and <span><span>[2]</span></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103607"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-05DOI: 10.1016/j.apal.2025.103606
Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist
We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality of a maximal cofinitary group (MCG) is strictly between and , and there is a -definable MCG of this cardinality. Here is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.
{"title":"Good projective witnesses","authors":"Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist","doi":"10.1016/j.apal.2025.103606","DOIUrl":"10.1016/j.apal.2025.103606","url":null,"abstract":"<div><div>We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality <span><math><msub><mrow><mi>a</mi></mrow><mrow><mtext>g</mtext></mrow></msub></math></span> of a maximal cofinitary group (MCG) is strictly between <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>c</mi></math></span>, and there is a <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-definable MCG of this cardinality. Here <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103606"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143927371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-05-02DOI: 10.1016/j.apal.2025.103605
Slavko Moconja , Predrag Tanović
We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type is weakly o-minimal if for some relatively A-definable linear order, <, on every relatively -definable subset of has finitely many convex components in . We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.
{"title":"Weakly o-minimal types","authors":"Slavko Moconja , Predrag Tanović","doi":"10.1016/j.apal.2025.103605","DOIUrl":"10.1016/j.apal.2025.103605","url":null,"abstract":"<div><div>We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type <span><math><mi>p</mi><mo>∈</mo><mi>S</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is weakly o-minimal if for some relatively <em>A</em>-definable linear order, <, on <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> every relatively <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span>-definable subset of <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> has finitely many convex components in <span><math><mo>(</mo><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>,</mo><mo><</mo><mo>)</mo></math></span>. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103605"},"PeriodicalIF":0.6,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-23DOI: 10.1016/j.apal.2025.103604
Derek Levinson
We find bounds for the maximal length of a sequence of distinct -sets under AD and show there is no sequence of distinct -sets of length . As a special case, there is no sequence of distinct -sets of length . These are the optimal results for the pointclasses and .
{"title":"Unreachability of Γ2n+1,m","authors":"Derek Levinson","doi":"10.1016/j.apal.2025.103604","DOIUrl":"10.1016/j.apal.2025.103604","url":null,"abstract":"<div><div>We find bounds for the maximal length of a sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets under <em>AD</em> and show there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span>-sets of length <span><math><msubsup><mrow><mi>δ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. As a special case, there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets of length <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>. These are the optimal results for the pointclasses <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103604"},"PeriodicalIF":0.6,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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