Pub Date : 2025-03-07DOI: 10.1016/j.apal.2025.103580
Wolfgang Rump
Main results on BL-algebras, including their classification in the finite case, are reconsidered and extended to a class of L-algebras X with prime factorization, including BL-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite BL-algebra reappears as a canonical L-subalgebra of prime elements in the self-similar closure where is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in X. These invariants determine X within its self-similar closure . The three basic types of BL-algebras are related to concepts of L-algebras with further-reaching significance in quantum theory.
{"title":"A complete invariant system for noetherian BL-algebras and more general L-algebras","authors":"Wolfgang Rump","doi":"10.1016/j.apal.2025.103580","DOIUrl":"10.1016/j.apal.2025.103580","url":null,"abstract":"<div><div>Main results on <em>BL</em>-algebras, including their classification in the finite case, are reconsidered and extended to a class of <em>L</em>-algebras <em>X</em> with prime factorization, including <em>BL</em>-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite <em>BL</em>-algebra reappears as a canonical <em>L</em>-subalgebra <span><math><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of prime elements in the self-similar closure <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> where <span><math><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in <em>X</em>. These invariants determine <em>X</em> within its self-similar closure <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>S</mi><mo>(</mo><mover><mrow><mi>P</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></math></span>. The three basic types of <em>BL</em>-algebras are related to concepts of <em>L</em>-algebras with further-reaching significance in quantum theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103580"},"PeriodicalIF":0.6,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620540","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 : 2025-03-04DOI: 10.1016/j.apal.2025.103570
Vera Fischer , L. Schembecker , David Schrittesser
We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of alongside the existence of a -well-order of the reals and a co-analytic witness for .
{"title":"Tight cofinitary groups","authors":"Vera Fischer , L. Schembecker , David Schrittesser","doi":"10.1016/j.apal.2025.103570","DOIUrl":"10.1016/j.apal.2025.103570","url":null,"abstract":"<div><div>We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mi>d</mi><mo><</mo><mi>c</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> alongside the existence of a <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-well-order of the reals and a co-analytic witness for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103570"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601168","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}
Pub Date : 2025-03-04DOI: 10.1016/j.apal.2025.103569
Morenikeji Neri
We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type p Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.
我们通过证明挖掘的证明论视角探索克罗内克两难的计算内容，并利用这一基本结果的有限变体，为在 p 型巴拿赫空间取值的随机变量的强大数定律提供新的速率，特别是在不依赖于随机变量分布的意义上，这种速率是非常均匀的。此外，我们还提供了可计算性理论论据来证明克罗内克∞的无效性，并从逆数学的角度研究了这一结果。此外，我们还提供了一个构造，说明可计算的收敛率并不总是可能的，以此证明克罗内克两难的无效性是如何向下渗透到大数强律的。最后，我们证明了克罗内克两难如何属于一类确定性公式，其辩证解释的解满足连续性属性，以及对于这类公式，我们如何获得一个升级原理，允许我们将确定性结果的计算解释提升为其概率类似的定量结果。这一结果概括了作者和皮施克之前的工作。
{"title":"A finitary Kronecker's lemma and large deviations in the strong law of large numbers on Banach spaces","authors":"Morenikeji Neri","doi":"10.1016/j.apal.2025.103569","DOIUrl":"10.1016/j.apal.2025.103569","url":null,"abstract":"<div><div>We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type <em>p</em> Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103569"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562528","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 : 2025-03-03DOI: 10.1016/j.apal.2025.103566
V. Fischer , L. Schembecker
We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.
We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under .
{"title":"Universally Sacks-indestructible combinatorial families of reals","authors":"V. Fischer , L. Schembecker","doi":"10.1016/j.apal.2025.103566","DOIUrl":"10.1016/j.apal.2025.103566","url":null,"abstract":"<div><div>We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.</div><div>We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing <span><math><msup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span> is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under <span><math><mi>CH</mi></math></span> we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under <span><math><mi>CH</mi></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103566"},"PeriodicalIF":0.6,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562527","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}
Pub Date : 2025-02-28DOI: 10.1016/j.apal.2025.103567
Miloš S. Kurilić
denotes the reduced power of a Boolean algebra , where Φ is the Fréchet filter on ω. We investigate iterated reduced powers ( and ) of collapsing algebras and our main intention is to classify the algebras , , up to isomorphism of their Boolean completions. In particular, assuming that SCH and hold, we show that for any cardinals and such that and we have , for each ; more precisely,
{"title":"Iterated reduced powers of collapsing algebras","authors":"Miloš S. Kurilić","doi":"10.1016/j.apal.2025.103567","DOIUrl":"10.1016/j.apal.2025.103567","url":null,"abstract":"<div><div><span><math><mrow><mi>rp</mi></mrow><mo>(</mo><mi>B</mi><mo>)</mo></math></span> denotes the reduced power <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>/</mo><mi>Φ</mi></math></span> of a Boolean algebra <span><math><mi>B</mi></math></span>, where Φ is the Fréchet filter on <em>ω</em>. We investigate iterated reduced powers (<span><math><msup><mrow><mi>rp</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi></math></span> and <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mrow><mi>rp</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>)</mo></math></span>) of collapsing algebras and our main intention is to classify the algebras <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, up to isomorphism of their Boolean completions. In particular, assuming that SCH and <span><math><mi>h</mi><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> hold, we show that for any cardinals <span><math><mi>λ</mi><mo>≥</mo><mi>ω</mi></math></span> and <span><math><mi>κ</mi><mo>≥</mo><mn>2</mn></math></span> such that <span><math><mi>κ</mi><mi>λ</mi><mo>></mo><mi>ω</mi></math></span> and <span><math><mrow><mi>cf</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>c</mi></math></span> we have <span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mo>(</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>, for each <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>; more precisely,<span><span><span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>c</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mtext> if </mtext><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>≤</mo><mi>c</mi><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspa","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103567"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547846","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 : 2025-02-28DOI: 10.1016/j.apal.2025.103568
Anand Pillay
We adapt the notion from [7] and [2] of a (relatively) definable subset of when M is a saturated structure, to the case when M is atomic and strongly ω-homogeneous (over a set A). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of . For example when T is stable, we have existence and uniqueness.
We also discuss the compatibility of our definability notions with definable Galois cohomology from [12] and differential Galois theory.
{"title":"Automorphism groups of prime models, and invariant measures","authors":"Anand Pillay","doi":"10.1016/j.apal.2025.103568","DOIUrl":"10.1016/j.apal.2025.103568","url":null,"abstract":"<div><div>We adapt the notion from <span><span>[7]</span></span> and <span><span>[2]</span></span> of a (relatively) definable subset of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> when <em>M</em> is a saturated structure, to the case <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span> when <em>M</em> is atomic and strongly <em>ω</em>-homogeneous (over a set <em>A</em>). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span>. For example when <em>T</em> is stable, we have existence and uniqueness.</div><div>We also discuss the compatibility of our definability notions with definable Galois cohomology from <span><span>[12]</span></span> and differential Galois theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103568"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547845","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}
Pub Date : 2025-02-21DOI: 10.1016/j.apal.2025.103565
Andrzej Rosłanowski , Saharon Shelah
We expand the results of Rosłanowski and Shelah [11], [10] to all perfect Abelian Polish groups . In particular, we show that if and , then there is a ccc forcing notion adding a set which has many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, imply that a translation of X or −X is included in Y.
{"title":"Borel sets without perfectly many overlapping translations, III","authors":"Andrzej Rosłanowski , Saharon Shelah","doi":"10.1016/j.apal.2025.103565","DOIUrl":"10.1016/j.apal.2025.103565","url":null,"abstract":"<div><div>We expand the results of Rosłanowski and Shelah <span><span>[11]</span></span>, <span><span>[10]</span></span> to all perfect Abelian Polish groups <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mo>+</mo><mo>)</mo></math></span>. In particular, we show that if <span><math><mi>α</mi><mo><</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>ω</mi></math></span>, then there is a ccc forcing notion adding a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> set <span><math><mi>B</mi><mo>⊆</mo><mi>H</mi></math></span> which has <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> many pairwise <em>k</em>–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, <span><math><mi>X</mi><mo>−</mo><mi>X</mi><mo>⊆</mo><mi>Y</mi><mo>−</mo><mi>Y</mi></math></span> imply that a translation of <em>X</em> or −<em>X</em> is included in <em>Y</em>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103565"},"PeriodicalIF":0.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547844","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 : 2025-02-18DOI: 10.1016/j.apal.2025.103564
Sapir Ben-Shahar , Heer Tern Koh
A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.
{"title":"Comparing notions of presentability in Polish spaces and Polish groups","authors":"Sapir Ben-Shahar , Heer Tern Koh","doi":"10.1016/j.apal.2025.103564","DOIUrl":"10.1016/j.apal.2025.103564","url":null,"abstract":"<div><div>A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103564"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463576","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}
Pub Date : 2025-02-17DOI: 10.1016/j.apal.2025.103563
Miguel Martins, Tommaso Moraschini
A bi-Heyting algebra validates the Gödel-Dummett axiom iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of is locally tabular.
Notably, if L is an axiomatic extension of , then L is locally tabular iff L is not contained in , the logic of a particular family of finite co-trees, called the finite combs. We prove that is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.
{"title":"Local tabularity is decidable for bi-intermediate logics of trees and of co-trees","authors":"Miguel Martins, Tommaso Moraschini","doi":"10.1016/j.apal.2025.103563","DOIUrl":"10.1016/j.apal.2025.103563","url":null,"abstract":"<div><div>A bi-Heyting algebra validates the Gödel-Dummett axiom <span><math><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> iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called <em>bi-Gödel algebras</em> and form a variety that algebraizes the extension <span><math><mi>bi-GD</mi></math></span> of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of <span><math><mi>bi-GD</mi></math></span> is locally tabular.</div><div>Notably, if <em>L</em> is an axiomatic extension of <span><math><mi>bi-GD</mi></math></span>, then <em>L</em> is locally tabular iff <em>L</em> is not contained in <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span>, the logic of a particular family of finite co-trees, called the <em>finite combs</em>. We prove that <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span> is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103563"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487726","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 : 2025-02-05DOI: 10.1016/j.apal.2025.103556
Sergio Celani , Rafał Gruszczyński , Paula Menchón
Drawing on the classic paper by Chellas [8], we propose a general algebraic framework for studying a binary operation of conditional that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of conditional algebras, and we develop its duality and canonical extensions theory.
{"title":"Conditional algebras","authors":"Sergio Celani , Rafał Gruszczyński , Paula Menchón","doi":"10.1016/j.apal.2025.103556","DOIUrl":"10.1016/j.apal.2025.103556","url":null,"abstract":"<div><div>Drawing on the classic paper by Chellas <span><span>[8]</span></span>, we propose a general algebraic framework for studying a binary operation of <em>conditional</em> that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of <em>conditional algebras</em>, and we develop its duality and canonical extensions theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103556"},"PeriodicalIF":0.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143376529","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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