{"title":"On Hausdorff operators in \u0000 \u0000 ZF\u0000 $mathsf {ZF}$","authors":"Kyriakos Keremedis, Eleftherios Tachtsis","doi":"10.1002/malq.202300004","DOIUrl":"https://doi.org/10.1002/malq.202300004","url":null,"abstract":"<p>A Hausdorff space <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>,</mo>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(X,mathcal {T})$</annotation>\u0000 </semantics></math> is called effectively Hausdorff if there exists a function <i>F</i>—called a Hausdorff operator—such that, for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>∈</mo>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation>$x,yin X$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>≠</mo>\u0000 <mi>y</mi>\u0000 </mrow>\u0000 <annotation>$xne y$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>F</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>(</mo>\u0000 <mi>U</mi>\u0000 <mo>,</mo>\u0000 <mi>V</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$F(x,y)=(U,V)$</annotation>\u0000 </semantics></math>, where <i>U</i> and <i>V</i> are disjoint open neighborhoods of <i>x</i> and <i>y</i>, respectively. Among other results, we establish the following in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (<math>\u0000 <semantics>\u0000 <mi>AC</mi>\u0000 <annotation>$mathsf {AC}$</annotation>\u0000 </semantics></math>):</p><p>\u0000 </p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 3","pages":"347-369"},"PeriodicalIF":0.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50142953","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":"Topological duality for orthomodular lattices","authors":"Joseph McDonald, Katalin Bimbó","doi":"10.1002/malq.202200044","DOIUrl":"https://doi.org/10.1002/malq.202200044","url":null,"abstract":"<p>A class of ordered relational topological spaces is described, which we call <i>orthomodular spaces</i>. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call <i>continuous weak p-morphisms</i>. It is well-known that orthomodular lattices provide an algebraic semantics for the quantum logic <math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathcal {Q}$</annotation>\u0000 </semantics></math>. Hence, as an application of our duality, we develop a topological semantics for <math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathcal {Q}$</annotation>\u0000 </semantics></math> using orthomodular spaces and prove soundness and completeness.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 2","pages":"174-191"},"PeriodicalIF":0.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50153949","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":"Bowtie-free graphs and generic automorphisms","authors":"Daoud Siniora","doi":"10.1002/malq.202200047","DOIUrl":"https://doi.org/10.1002/malq.202200047","url":null,"abstract":"<p>We show that the countable universal ω-categorical bowtie-free graph admits generic automorphisms. Moreover, we show that this graph is not finitely homogenisable.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 2","pages":"221-230"},"PeriodicalIF":0.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50119031","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":"Avoiding Medvedev reductions inside a linear order","authors":"Noah Schweber","doi":"10.1002/malq.202200059","DOIUrl":"https://doi.org/10.1002/malq.202200059","url":null,"abstract":"<p>While every endpointed interval <i>I</i> in a linear order <i>J</i> is, considered as a linear order in its own right, trivially Muchnik-reducible to <i>J</i> itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 2","pages":"165-173"},"PeriodicalIF":0.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50153950","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 categorical equivalence between logical quantale modules and quantum B-modules","authors":"Xianglong Ruan, Xiaochuan Liu","doi":"10.1002/malq.202200062","DOIUrl":"10.1002/malq.202200062","url":null,"abstract":"<p>This paper introduces the notion of logical quantale module. It proves that there is a dual equivalence between the category of logical quantale modules and the category of quantum B-modules, in the way that every quantum B-module admits a natural embedding into a logical quantale module, the enveloping quantale module.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 4","pages":"446-459"},"PeriodicalIF":0.3,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84190356","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":"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":"69 2","pages":"147-164"},"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":"69 2","pages":"207-220"},"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":"69 1","pages":"6"},"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":"69 1","pages":"92-103"},"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}
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