{"title":"Countable ordered groups and Weihrauch reducibility","authors":"Ang Li","doi":"10.1016/j.apal.2025.103644","DOIUrl":"10.1016/j.apal.2025.103644","url":null,"abstract":"<div><div>This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem <span><span>[11]</span></span> on the order types of countable ordered groups. Solomon <span><span>[14]</span></span> showed that the theorem is equivalent to <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-<span><math><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, the strongest of the big five subsystems of second order arithmetic. We show that the strength of the theorem comes from having a dense linear order without endpoints in its order type. Then, we show that for the related Weihrauch problem to be strong enough to be equivalent to <span><math><mover><mrow><mi>WF</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> (the analog problem of <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-<span><math><mi>C</mi><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>), an order-preserving function is necessary in the output. Without the order-preserving function, the problems are very much to the side compared to analog problems of the big five.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103644"},"PeriodicalIF":0.6,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771199","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}
{"title":"Thick sets and the Central Set Theorem","authors":"Teng Zhang","doi":"10.1016/j.apal.2025.103645","DOIUrl":"10.1016/j.apal.2025.103645","url":null,"abstract":"<div><div>In 1981, Furstenberg introduced the notion of central sets, and he established the Central Set Theorem. Since then, several generalizations of this result have been found, where a significant version is obtained by De, Hindman and Strauss. In this article, we find that the Central Set Theorem can be improved further. And we observe that there are some connections between thick sets and <em>J</em>-sets. Based on that, we establish a CST-type result for thick sets. Moreover, we introduce a new notion called super thick sets, and find that this notion has rich combinatorial properties. In particular, it contains additive and multiplicative structures, and it has a CST-type result for two operations. In addition, it can be partitioned into <em>κ</em> super thick subsets in very weakly cancellative weak rings with size <em>κ</em>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103645"},"PeriodicalIF":0.6,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144723034","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}
{"title":"Description of structures computable in polynomial time","authors":"Pavel E. Alaev","doi":"10.1016/j.apal.2025.103636","DOIUrl":"10.1016/j.apal.2025.103636","url":null,"abstract":"<div><div>We describe structures computable in polynomial time (P-computable) by a simple and short criterion. We prove that every substructure of a P-computable structure generated by a c.e. set also has a P-computable presentation. For example, we easily prove that every computable torsion-free Abelian group has a P-computable presentation.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103636"},"PeriodicalIF":0.6,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144703217","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}
{"title":"Unified inverse correspondence for LE-logics","authors":"Alessandra Palmigiano , Mattia Panettiere","doi":"10.1016/j.apal.2025.103635","DOIUrl":"10.1016/j.apal.2025.103635","url":null,"abstract":"<div><div>We generalize Kracht's theory of internal describability from classical modal logic to the family of all logics canonically associated with varieties of normal lattice expansions (LE algebras). We work in the purely algebraic setting of perfect LEs; the formulas playing the role of Kracht's formulas in this generalized setting pertain to a first order language whose atoms are special inequalities between terms of perfect algebras. Via duality, formulas in this language can be equivalently translated into first order conditions in the frame correspondence languages of several types of relational semantics for LE-logics.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 1","pages":"Article 103635"},"PeriodicalIF":0.6,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694400","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}
{"title":"The DeMorganization of a locale","authors":"Igor Arrieta","doi":"10.1016/j.apal.2025.103634","DOIUrl":"10.1016/j.apal.2025.103634","url":null,"abstract":"<div><div>In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103634"},"PeriodicalIF":0.6,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587704","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}
{"title":"The unstable formula theorem revisited via algorithms","authors":"M. Malliaris , S. Moran","doi":"10.1016/j.apal.2025.103633","DOIUrl":"10.1016/j.apal.2025.103633","url":null,"abstract":"<div><div>This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called “Probably Eventually Correct” or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103633"},"PeriodicalIF":0.6,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632237","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}
{"title":"On definable Skolem functions and trichotomy","authors":"Bruno Dinis , Mário J. Edmundo","doi":"10.1016/j.apal.2025.103632","DOIUrl":"10.1016/j.apal.2025.103632","url":null,"abstract":"<div><div>In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over ∅ and finitely many open intervals each a union of a ∅-definable family of group-intervals with fixed positive elements.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103632"},"PeriodicalIF":0.6,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580943","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}
{"title":"On Recurrence Axioms","authors":"Sakaé Fuchino , Toshimichi Usuba","doi":"10.1016/j.apal.2025.103631","DOIUrl":"10.1016/j.apal.2025.103631","url":null,"abstract":"<div><div>The Recurrence Axiom for a class <span><math><mi>P</mi></math></span> of posets and a set <em>A</em> of parameters is an axiom scheme in the language of <span>ZFC</span> asserting that if a statement with parameters from <em>A</em> is forced by a poset in <span><math><mi>P</mi></math></span>, then there is a ground containing the parameters and satisfying the statement.</div><div>The tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge continuum implies the Recurrence Axiom for <span><math><mi>P</mi></math></span> and <span><math><mi>H</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>)</mo></math></span>. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly <span><math><mi>P</mi></math></span>-generic hyperhuge cardinal <em>κ</em>, and that <em>κ</em> in the bedrock is genuinely hyperhuge, or even super <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span> hyperhuge if <em>κ</em> is a tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge definable cardinal.</div><div>The Laver Generic Maximum (<span>LGM</span>), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of <span>LGM</span> while Cichoń's Maximum is a phenomenon in many grounds of the universe under <span>LGM</span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103631"},"PeriodicalIF":0.6,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144556730","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}
{"title":"Topological games in Ramsey spaces","authors":"Julián C. Cano , Carlos A. Di Prisco","doi":"10.1016/j.apal.2025.103630","DOIUrl":"10.1016/j.apal.2025.103630","url":null,"abstract":"<div><div>Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103630"},"PeriodicalIF":0.6,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514036","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}
{"title":"Unstable independence from the categorical point of view","authors":"Mark Kamsma , Jiří Rosický","doi":"10.1016/j.apal.2025.103629","DOIUrl":"10.1016/j.apal.2025.103629","url":null,"abstract":"<div><div>We give a category-theoretic construction of simple and NSOP<sub>1</sub>-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying properties of a class of monomorphisms <span><math><mi>M</mi></math></span> such that the pullback squares consisting of morphisms in <span><math><mi>M</mi></math></span> form the desired independence relation. This generalizes the category-theoretic construction of stable independence relations using effective unions or cellular squares by M. Lieberman, S. Vasey and the second author to the unstable setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103629"},"PeriodicalIF":0.6,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322319","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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