{"title":"Topological properties of definable sets in ordered Abelian groups of burden 2","authors":"Alfred Dolich, John Goodrick","doi":"10.1002/malq.202200052","DOIUrl":"https://doi.org/10.1002/malq.202200052","url":null,"abstract":"<p>We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp-rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle-third set (Theorem 2.9). If it has burden 2 and both an infinite discrete set <i>D</i> and a dense-codense set <i>X</i> are definable, then translates of <i>X</i> must witness the Independence Property (Theorem 2.26). In the last section, an explicit example of an ordered Abelian group of burden 2 is given in which both an infinite discrete set and a dense-codense set are definable.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50132671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the variety of strong subresiduated lattices","authors":"Sergio Celani, Hernán J. San Martín","doi":"10.1002/malq.202200067","DOIUrl":"https://doi.org/10.1002/malq.202200067","url":null,"abstract":"<p>A subresiduated lattice is a pair <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>,</mo>\u0000 <mi>D</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(A,D)$</annotation>\u0000 </semantics></math>, where <i>A</i> is a bounded distributive lattice, <i>D</i> is a bounded sublattice of <i>A</i> and for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 <mo>∈</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <annotation>$a,bin A$</annotation>\u0000 </semantics></math> there exists the maximum of the set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mi>d</mi>\u0000 <mo>∈</mo>\u0000 <mi>D</mi>\u0000 <mo>:</mo>\u0000 <mi>a</mi>\u0000 <mo>∧</mo>\u0000 <mi>d</mi>\u0000 <mo>≤</mo>\u0000 <mi>b</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$lbrace din D:awedge dle brbrace$</annotation>\u0000 </semantics></math>, which is denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>→</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$arightarrow b$</annotation>\u0000 </semantics></math>. This pair can be regarded as an algebra <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>,</mo>\u0000 <mo>∧</mo>\u0000 <mo>,</mo>\u0000 <mo>∨</mo>\u0000 <mo>,</mo>\u0000 <mo>→</mo>\u0000 <mo>,</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(A,wedge ,vee ,rightarrow ,0,1)$</annotation>\u0000 </semantics></math> of type (2, 2, 2, 0, 0), where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>=</mo>\u0000 <mo>{</mo>\u0000 <mi>a</mi>\u0000 <mo>∈</mo>\u0000 <mi>A</mi>\u0000 <mo>:</mo>\u0000 <mn>1</mn>\u0000 <mo>→</mo>\u0000 <mi>a</mi>\u0000 <mo>=</mo>\u0000 <mi>a</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$D=lbrace ain A: 1rightarrow a =arbrace$</annotation>\u0000 </semantics></math>. The class of subresiduated lattices is a variety which properly contains the variety of Heyting algebras. In this paper we study the subvariety of subresiduated lattices, denoted by <math>\u0000 <semantics>\u0000 <msup>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50128774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"","authors":"","doi":"10.1002/malq.202310002","DOIUrl":"https://doi.org/10.1002/malq.202310002","url":null,"abstract":"<p>Dear Readers,</p><p>We wish all of you a happy and successful year 2023.</p><p>The issue you are looking at presents our journal in a new layout, the new standardised journal design used by our publisher Wiley-VCH Verlag. Apart from the major changes on the title page of articles, most of the features of the layout of the journal Mathematical Logic Quarterly (MLQ) were retained.</p><p>But the appearance of the papers is not the only major change at MLQ. After having served for a dozen years as one of the Managing Editors of MLQ, Benedikt Löwe has decided to step down at the end of 2022. In 2011, the journal underwent some significant changes as it started to be published under the auspices of the Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG). Benedikt served as vice president (until 2012) and later as president of the DVMLG (2012–2022) and directed this transition forcefully.</p><p>In addition to being one of the Managing Editors, Benedikt served as the liaison between the DVMLG, the Editorial Office, the publisher, and the typesetters; he worked closely with our past Editorial Assistants, Peter van Ormondt and Hugo Nobrega, as well as our current Editorial Manager, Thomas Piecha. Benedikt's clear and pronounced ideas about scientific publishing have always been appreciated and certainly significantly formed MLQ's current character. We shall try to keep his clear views on scientific publishing in mind. Thank you, Benedikt, for 12 years of inspiring and productive collaboration.</p><p>We are happy that Manuel Bodirsky from the Technische Universität Dresden will replace Benedikt both as Managing Editor and as representative of the DVMLG's Board in the Editorial Office of MLQ. Manuel is well known to us and our authors as he has been a member of the Editorial Board of MLQ since 2017. His main fields of research are Constraint Satisfaction Problems and Universal Algebra. Welcome to the Editorial Office, Manuel!</p><p>Since our last editorial was published in Volume 66 (2020), a number of members of the Editorial Board were re-appointed upon nomination by the Board of the DVMLG for another term of office for three years: Steve Awodey, Zoé Chatzidakis, Victoria Gitman, Andrew Marks, and Alexandra Silva for an additional term from 1 January 2021 to 31 December 2023; Nick Bezhanishvili, Su Gao, Maria Emilia Maietti, and Anush Tserunyan for an additional term from 1 January 2022 to 31 December 2024; and John Baldwin, Artem Chernikov, Rod Downey, Ilijas Farah, Ekaterina Fokina, Stefan Geschke, Hajime Ishihara, Franziska Jahnke, and Dima Sinapova for an additional term from 1 January 2023 to 31 December 2025. After nine years of service in the Editorial Board, Jan Krajíček left the board at the end of 2022. Thanks to Jan for his commitment to make MLQ a successful project.</p><p>As new members of the Editorial Board, the following researchers were appointed for a first term from 1 January 202","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202310002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cofinal types on ω2","authors":"Borisa Kuzeljevic, Stevo Todorcevic","doi":"10.1002/malq.202200021","DOIUrl":"https://doi.org/10.1002/malq.202200021","url":null,"abstract":"<p>In this paper we start the analysis of the class <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$mathcal {D}_{aleph _2}$</annotation>\u0000 </semantics></math>, the class of cofinal types of directed sets of cofinality at most ℵ<sub>2</sub>. We compare elements of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$mathcal {D}_{aleph _2}$</annotation>\u0000 </semantics></math> using the notion of Tukey reducibility. We isolate some simple cofinal types in <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$mathcal {D}_{aleph _2}$</annotation>\u0000 </semantics></math>, and then proceed to find some of these types which have an immediate successor in the Tukey ordering of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$mathcal {D}_{aleph _2}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50149486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A proof-theoretic metatheorem for tracial von Neumann algebras","authors":"Liviu Păunescu, Andrei Sipoş","doi":"10.1002/malq.202200048","DOIUrl":"https://doi.org/10.1002/malq.202200048","url":null,"abstract":"<p>We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research programme that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Nice ℵ1 generated non-P-points, Part I","authors":"Saharon Shelah","doi":"10.1002/malq.202200070","DOIUrl":"https://doi.org/10.1002/malq.202200070","url":null,"abstract":"<p>We define a family of non-principal ultrafilters on <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>${mathbb {N}}$</annotation>\u0000 </semantics></math> which are, in a sense, very far from P-points. We prove the existence of such ultrafilters under reasonable conditions. In subsequent articles, we intend to prove that such ultrafilters may exist while no P-point exists. Though our primary motivations came from forcing and independence results, the family of ultrafilters introduced here should be interesting from combinatorial point of view too.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200070","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The cofinality of the strong measure zero ideal for κ inaccessible","authors":"Johannes Philipp Schürz","doi":"10.1002/malq.202000093","DOIUrl":"https://doi.org/10.1002/malq.202000093","url":null,"abstract":"<p>We investigate the cofinality of the strong measure zero ideal for κ inaccessible and show that it is independent of the size of 2<sup>κ</sup>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000093","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some definable types that cannot be amalgamated","authors":"Martin Hils, Rosario Mennuni","doi":"10.1002/malq.202200046","DOIUrl":"https://doi.org/10.1002/malq.202200046","url":null,"abstract":"<p>We exhibit a theory where definable types lack the amalgamation property.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The subset relation and 2-stratified sentences in set theory and class theory","authors":"Zachiri McKenzie","doi":"10.1002/malq.202200029","DOIUrl":"https://doi.org/10.1002/malq.202200029","url":null,"abstract":"<p>Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mi>BAS</mi>\u0000 <annotation>$mathsf {BAS}$</annotation>\u0000 </semantics></math>, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a complete extension, <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>IABA</mi>\u0000 <mi>Ideal</mi>\u0000 </msub>\u0000 <annotation>$mathsf {IABA}_{mathsf {Ideal}}$</annotation>\u0000 </semantics></math>, of the theory of infinite atomic boolean algebras and a minimum subsystem, <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>BAC</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$mathsf {BAC}^+$</annotation>\u0000 </semantics></math>, of <math>\u0000 <semantics>\u0000 <mi>NBG</mi>\u0000 <annotation>$mathsf {NBG}$</annotation>\u0000 </semantics></math> are identified with the property that if <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> is a model of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>BAC</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$mathsf {BAC}^+$</annotation>\u0000 </semantics></math>, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>⟨</mo>\u0000 <mi>M</mi>\u0000 <mo>,</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mi>M</mi>\u0000 </msup>\u0000 <mo>,</mo>\u0000 <msup>\u0000 <mo>⊆</mo>\u0000 <mi>M</mi>\u0000 </msup>\u0000 <mo>⟩</mo>\u0000 </mrow>\u0000 <annotation>$langle M, mathcal {S}^mathcal {M}, subseteq ^mathcal {M} rangle$</annotation>\u0000 </semantics></math> is a model of <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>IABA</mi>\u0000 <mi>Ideal</mi>\u0000 </msub>\u0000 <annotation>$mathsf {IABA}_{mathsf {Ideal}}$</annotation>\u0000 </semantics></math>, where <i>M</i> is the underlying set of <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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