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Positive definability patterns 正可定义性模式
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2024-11-28 DOI: 10.1016/j.apal.2024.103539
Ori Segel
{"title":"Positive definability patterns","authors":"Ori Segel","doi":"10.1016/j.apal.2024.103539","DOIUrl":"10.1016/j.apal.2024.103539","url":null,"abstract":"<div><div>We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory <em>T</em> we put two structures on the type spaces of models of <em>T</em> in two languages, <span><math><mi>L</mi></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span>. It turns out that for sufficiently saturated models, the corresponding h-universal theories <span><math><mi>T</mi></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> are independent of the model. We show that there is a canonical model <span><math><mi>J</mi></math></span> of <span><math><mi>T</mi></math></span>, and in many interesting cases there is an analogous canonical model <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span>, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103539"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162349","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
Ordered transexponential fields 有序转幂域
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2024-12-02 DOI: 10.1016/j.apal.2024.103541
Lothar Sebastian Krapp , Salma Kuhlmann
{"title":"Ordered transexponential fields","authors":"Lothar Sebastian Krapp ,&nbsp;Salma Kuhlmann","doi":"10.1016/j.apal.2024.103541","DOIUrl":"10.1016/j.apal.2024.103541","url":null,"abstract":"<div><div>We develop a first-order theory of ordered transexponential fields in the language <span><math><mo>{</mo><mo>+</mo><mo>,</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>&lt;</mo><mo>,</mo><mi>e</mi><mo>,</mo><mi>T</mi><mo>}</mo></math></span>, where <em>e</em> and <em>T</em> stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>e</mi><mo>)</mo></math></span> to admit a transexponential function <em>T</em> compatible with <em>e</em>. Moreover, we give a full characterisation of all <em>countable</em> ordered transexponential fields in terms of their valuation theoretic invariants.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103541"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162350","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
Tame topology in Hensel minimal structures Hensel最小结构中的驯服拓扑
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2024-11-29 DOI: 10.1016/j.apal.2024.103540
Krzysztof Jan Nowak
{"title":"Tame topology in Hensel minimal structures","authors":"Krzysztof Jan Nowak","doi":"10.1016/j.apal.2024.103540","DOIUrl":"10.1016/j.apal.2024.103540","url":null,"abstract":"<div><div>We are concerned with topology of Hensel minimal structures on non-trivially valued fields <em>K</em>, whose axiomatic theory was introduced in a recent paper by Cluckers–Halupczok–Rideau. We additionally require that every definable subset in the imaginary sort <em>RV</em>, binding together the residue field <em>Kv</em> and value group <em>vK</em>, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets. These are, among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Łojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103540"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162351","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
Π2-rule systems and inductive classes of Gödel algebras Gödel代数的Π2-rule系统和归纳类
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2025-01-14 DOI: 10.1016/j.apal.2025.103552
Rodrigo Nicolau Almeida
{"title":"Π2-rule systems and inductive classes of Gödel algebras","authors":"Rodrigo Nicolau Almeida","doi":"10.1016/j.apal.2025.103552","DOIUrl":"10.1016/j.apal.2025.103552","url":null,"abstract":"<div><div>In this paper we present a general theory of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules for systems of intuitionistic and modal logic. We introduce the notions of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule systems extending <span><math><mrow><mi>LC</mi></mrow><mo>=</mo><mrow><mi>IPC</mi></mrow><mo>+</mo><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math></span>, and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in <span><math><mi>LC</mi></math></span>: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules which are admissible are derivable, and (2) show that the problem of admissibility of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules over <span><math><mi>LC</mi></math></span> is decidable.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103552"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Modal logics over lattices 格上的模态逻辑
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2025-01-13 DOI: 10.1016/j.apal.2025.103553
Xiaoyang Wang , Yanjing Wang
{"title":"Modal logics over lattices","authors":"Xiaoyang Wang ,&nbsp;Yanjing Wang","doi":"10.1016/j.apal.2025.103553","DOIUrl":"10.1016/j.apal.2025.103553","url":null,"abstract":"<div><div>Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) <span><span>[8]</span></span>. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of <em>infimum</em> and <em>supremum</em> can reveal more structures behind various lattices.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103553"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162323","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
The logic of cardinality comparison without the axiom of choice 没有选择公理的基数比较逻辑
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2024-12-10 DOI: 10.1016/j.apal.2024.103549
Matthew Harrison-Trainor , Dhruv Kulshreshtha
{"title":"The logic of cardinality comparison without the axiom of choice","authors":"Matthew Harrison-Trainor ,&nbsp;Dhruv Kulshreshtha","doi":"10.1016/j.apal.2024.103549","DOIUrl":"10.1016/j.apal.2024.103549","url":null,"abstract":"<div><div>We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.</div><div>A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-<em>m</em>). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103549"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162346","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 dp-minimal expansions of the integers 关于整数的dp-极小展开
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-04-01 Epub Date: 2025-01-09 DOI: 10.1016/j.apal.2024.103551
Eran Alouf
{"title":"On dp-minimal expansions of the integers","authors":"Eran Alouf","doi":"10.1016/j.apal.2024.103551","DOIUrl":"10.1016/j.apal.2024.103551","url":null,"abstract":"<div><div>We show that if <span><math><mi>Z</mi></math></span> is a dp-minimal expansion of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> that defines an infinite subset of <span><math><mi>N</mi></math></span>, then <span><math><mi>Z</mi></math></span> is interdefinable with <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>&lt;</mo><mo>)</mo></math></span>. As a corollary, we show the same for dp-minimal expansions of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> which do not eliminate <span><math><msup><mrow><mo>∃</mo></mrow><mrow><mo>∞</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103551"},"PeriodicalIF":0.6,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162324","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
Strong standard completeness theorems for S5-modal Łukasiewicz logics 5-模态Łukasiewicz逻辑的强标准完备性定理
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-03-01 Epub Date: 2024-11-22 DOI: 10.1016/j.apal.2024.103529
Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy
{"title":"Strong standard completeness theorems for S5-modal Łukasiewicz logics","authors":"Diego Castaño ,&nbsp;José Patricio Díaz Varela ,&nbsp;Gabriel Savoy","doi":"10.1016/j.apal.2024.103529","DOIUrl":"10.1016/j.apal.2024.103529","url":null,"abstract":"<div><div>We study the S5-modal expansion of the Łukasiewicz logic. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MV-algebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 3","pages":"Article 103529"},"PeriodicalIF":0.6,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746012","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
Semiconic idempotent logic II: Beth definability and deductive interpolation 半符号幂等逻辑II：贝丝可定义性与演绎插值
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-03-01 Epub Date: 2024-11-15 DOI: 10.1016/j.apal.2024.103528
Wesley Fussner , Nikolaos Galatos
{"title":"Semiconic idempotent logic II: Beth definability and deductive interpolation","authors":"Wesley Fussner ,&nbsp;Nikolaos Galatos","doi":"10.1016/j.apal.2024.103528","DOIUrl":"10.1016/j.apal.2024.103528","url":null,"abstract":"<div><div>Semiconic idempotent logic <strong>sCI</strong> is a common generalization of intuitionistic logic, semilinear idempotent logic <strong>sLI</strong>, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of <strong>sCI</strong>, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of stratified and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by stratified and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of <strong>sCI</strong>, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending <strong>sCI</strong>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 3","pages":"Article 103528"},"PeriodicalIF":0.6,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143148056","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
Saturation properties for compositional truth with propositional correctness 具有命题正确性的组合真理的饱和特性
IF 0.6 2区 数学
Annals of Pure and Applied Logic Pub Date : 2025-02-01 Epub Date: 2024-09-03 DOI: 10.1016/j.apal.2024.103512
Bartosz Wcisło
{"title":"Saturation properties for compositional truth with propositional correctness","authors":"Bartosz Wcisło","doi":"10.1016/j.apal.2024.103512","DOIUrl":"10.1016/j.apal.2024.103512","url":null,"abstract":"<div><p>It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103512"},"PeriodicalIF":0.6,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001167/pdfft?md5=93f2e704b024dfc73e7a30a7ab95c178&pid=1-s2.0-S0168007224001167-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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