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Equational definitions of logical filters 逻辑滤波器的等式定义
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-28 DOI: 10.1016/j.apal.2025.103617
Michele Pra Baldi , Adam Přenosil
{"title":"Equational definitions of logical filters","authors":"Michele Pra Baldi ,&nbsp;Adam Přenosil","doi":"10.1016/j.apal.2025.103617","DOIUrl":"10.1016/j.apal.2025.103617","url":null,"abstract":"<div><div>A finitary propositional logic can be given an algebraic reading in two different ways: by translating formulas into equations and logical rules into quasi-equations, or by translating logical rules directly into equations. The former type of algebraic interpretation has been extensively studied and underlies the theory of algebraization, whereas little systematic attention has been paid to the latter type. We investigate a semantic form of the latter type of algebraic interpretation, which we call the equational definability of compact filters (EDCF). Paralleling the well-studied hierarchy of variants of the deduction–detachment theorem (DDT), this property also comes in local, parametrized, and parametrized local variants. The main results of this paper characterize of each of these variants of the EDCF in a spirit similar to the existing characterizations of the DDT. While the EDCF hierarchy and the DDT hierarchy coincide for algebraizable logics, part of the interest of the EDCF stems from the fact it is often enjoyed even by logics which are not well-behaved in terms of other existing classifications in algebraic logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103617"},"PeriodicalIF":0.6,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144170698","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}
引用次数: 0
Interpolation in Hájek's basic logic 插值在Hájek的基本逻辑
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-26 DOI: 10.1016/j.apal.2025.103615
Wesley Fussner , Simon Santschi
{"title":"Interpolation in Hájek's basic logic","authors":"Wesley Fussner ,&nbsp;Simon Santschi","doi":"10.1016/j.apal.2025.103615","DOIUrl":"10.1016/j.apal.2025.103615","url":null,"abstract":"<div><div>We exhaustively classify varieties of BL-algebras with the amalgamation property, showing that there are only countably many of them and solving an open problem of Montagna. As a consequence of this classification, we obtain a complete description of which axiomatic extensions of Hájek's basic fuzzy logic <strong>BL</strong> have the deductive interpolation property. Along the way, we provide similar classifications of varieties of basic hoops with the amalgamation property and axiomatic extensions of the negation-free fragment of <strong>BL</strong> with the deductive interpolation property.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103615"},"PeriodicalIF":0.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147947","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}
引用次数: 0
Model-theoretic K1 of free modules over PIDs pid上自由模的模型论K1
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-22 DOI: 10.1016/j.apal.2025.103613
Sourayan Banerjee, Amit Kuber
{"title":"Model-theoretic K1 of free modules over PIDs","authors":"Sourayan Banerjee,&nbsp;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}
引用次数: 0
Descriptive complexity of topological invariants 拓扑不变量的描述复杂性
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-14 DOI: 10.1016/j.apal.2025.103611
Djamel Eddine Amir, Mathieu Hoyrup
{"title":"Descriptive complexity of topological invariants","authors":"Djamel Eddine Amir,&nbsp;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}
引用次数: 0
Generically computable linear orderings
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-12 DOI: 10.1016/j.apal.2025.103612
Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov
{"title":"Generically computable linear orderings","authors":"Wesley Calvert ,&nbsp;Douglas Cenzer ,&nbsp;David Gonzalez ,&nbsp;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}
引用次数: 0
Spaces not distinguishing ideal pointwise and σ-uniform convergence
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-08 DOI: 10.1016/j.apal.2025.103609
Rafał Filipów, Adam Kwela
{"title":"Spaces not distinguishing ideal pointwise and σ-uniform convergence","authors":"Rafał Filipów,&nbsp;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}
引用次数: 0
Free p-algebras revisited: An algebraic investigation of implication-free intuitionism
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-08 DOI: 10.1016/j.apal.2025.103610
Tomasz Kowalski , Katarzyna Słomczyńska
{"title":"Free p-algebras revisited: An algebraic investigation of implication-free intuitionism","authors":"Tomasz Kowalski ,&nbsp;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}
引用次数: 0
On cardinal invariants related to Rosenthal families and large-scale topology 关于Rosenthal族与大尺度拓扑的基数不变量
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-05 DOI: 10.1016/j.apal.2025.103607
Arturo Martínez-Celis, Tomasz Żuchowski
{"title":"On cardinal invariants related to Rosenthal families and large-scale topology","authors":"Arturo Martínez-Celis,&nbsp;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}
引用次数: 0
Good projective witnesses 好的投影证人
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-05 DOI: 10.1016/j.apal.2025.103606
Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist
{"title":"Good projective witnesses","authors":"Vera Fischer ,&nbsp;Sy David Friedman ,&nbsp;David Schrittesser ,&nbsp;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}
引用次数: 0
Weakly o-minimal types 弱o极小型
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-05-02 DOI: 10.1016/j.apal.2025.103605
Slavko Moconja , Predrag Tanović
{"title":"Weakly o-minimal types","authors":"Slavko Moconja ,&nbsp;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, &lt;, 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>&lt;</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}
引用次数: 0
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