Mathematical Logic Quarterly最新文献

{"title":"Wadge degrees of \u0000 \u0000 \u0000 Δ\u0000 2\u0000 0\u0000 \u0000 $mathbf{Delta }^0_2$\u0000 omega-powers","authors":"Olivier Finkel, Dominique Lecomte","doi":"10.1002/malq.202400024","DOIUrl":"10.1002/malq.202400024","url":null,"abstract":"<p>We provide, for each natural number <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> and each class among <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$D_n(mathbf {Sigma }^0_1)$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mover>\u0000 <mi>D</mi>\u0000 <mo>̌</mo>\u0000 </mover>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$check{D}_n(mathbf {Sigma }^0_1)$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>⊕</mi>\u0000 <msub>\u0000 <mover>\u0000 <mi>D</mi>\u0000 <mo>̌</mo>\u0000 </mover>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mn>1</mn>\u0000 <mn>0</mn>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$D_{2n+1}(mathbf {Sigma }^0_1)oplus check{D}_{2n+1}(mathbf {Sigma }^0_1)$</annotation>\u0000 </semantics></math>, a regular language whose ass","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"286-293"},"PeriodicalIF":0.4,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182743","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":"Contents: (Math. Log. Quart. 2/2024)","authors":"","doi":"10.1002/malq.202470022","DOIUrl":"10.1002/malq.202470022","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"139-141"},"PeriodicalIF":0.4,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202470022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770342","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":"Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups","authors":"Eliana Barriga","doi":"10.1002/malq.202300028","DOIUrl":"10.1002/malq.202300028","url":null,"abstract":"<p>We state conditions for which a definable local homomorphism between two locally definable groups <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$mathcal {G}$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>G</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <annotation>$mathcal {G^{prime }}$</annotation>\u0000 </semantics></math> can be uniquely extended when <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$mathcal {G}$</annotation>\u0000 </semantics></math> is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> not necessarily abelian over a sufficiently saturated real closed field <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>; namely, that the o-minimal universal covering group <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>G</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <annotation>$widetilde{G}$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is an open locally definable subgroup of <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mn>0</mn>\u0000 </msup>\u0000 </mrow>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <annotation>$widetilde{H(R)^{0}}$</annotation>\u0000 </semantics></math> for some <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-algebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"267-274"},"PeriodicalIF":0.4,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745481","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 Hartogs–Lindenbaum spectrum of symmetric extensions","authors":"Calliope Ryan-Smith","doi":"10.1002/malq.202300047","DOIUrl":"10.1002/malq.202300047","url":null,"abstract":"<p>We expand the classic result that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>AC</mi>\u0000 <mi>WO</mi>\u0000 </msub>\u0000 <annotation>$mathsf {AC}_mathsf {WO}$</annotation>\u0000 </semantics></math> is equivalent to the statement “For all <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℵ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>ℵ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$aleph (X)=aleph ^*(X)$</annotation>\u0000 </semantics></math>” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of <span></span><math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of <span></span><math>\u0000 <semantics>\u0000 <mi>ZFC</mi>\u0000 <annotation>$mathsf {ZFC}$</annotation>\u0000 </semantics></math>. We prove that all such spectra fall into a very rigid pattern.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"210-223"},"PeriodicalIF":0.4,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720944","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":"Filter-Menger set of reals in Cohen extensions","authors":"Hang Zhang,&nbsp;Shuguo Zhang","doi":"10.1002/malq.202300008","DOIUrl":"10.1002/malq.202300008","url":null,"abstract":"&lt;p&gt;We prove that for every ultrafilter &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {U}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;annotation&gt;$omega$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; there exists a filter &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {F}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$2^{&amp;lt;omega }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which is &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {U}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Menger and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;χ&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;U&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$chi (mathcal {F})=mathfrak {b}(mathcal {U})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We show that in the Cohen model there exists such &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {F}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathfrak {d}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; that is not Hurewicz in the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;annotation&gt;$kappa$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-Cohen model where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;κ&lt;/mi&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ω&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$kappa &amp;gt;omega _{1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;b&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathfrak {b}&amp;lt;mathfrak {d}$&lt;/annotation&gt;\u0000 &lt;/se","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"224-232"},"PeriodicalIF":0.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141645034","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":"Expansions of real closed fields with the Banach fixed point property","authors":"Athipat Thamrongthanyalak","doi":"10.1002/malq.202400001","DOIUrl":"10.1002/malq.202400001","url":null,"abstract":"<p>We study a variant of converses of the Banach fixed point theorem and its connection to tameness in expansions of a real closed field. An expansion of a real closed ordered field is said to have the Banach fixed point property when, for every locally closed definable set <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math>, if every definable contraction on <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> has a fixed point, then <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> is closed. Let <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathfrak {R}$</annotation>\u0000 </semantics></math> be an expansion of a real closed field. We prove that if <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathfrak {R}$</annotation>\u0000 </semantics></math> has an o-minimal open core, then it has the Banach fixed point property; and if <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathfrak {R}$</annotation>\u0000 </semantics></math> is definably complete and has the Banach fixed point property, then it has a locally o-minimal open core.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"197-204"},"PeriodicalIF":0.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141646827","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":"Hilbert's tenth problem for lacunary entire functions of finite order","authors":"Natalia Garcia-Fritz,&nbsp;Hector Pasten","doi":"10.1002/malq.202300046","DOIUrl":"10.1002/malq.202300046","url":null,"abstract":"<p>In the context of Hilbert's tenth problem, an outstanding open case is that of complex entire functions in one variable. A negative solution is known for polynomials (by Denef) and for exponential polynomials of finite order (by Chompitaki, Garcia-Fritz, Pasten, Pheidas, and Vidaux), but no other case is known for rings of complex entire functions in one variable. We prove a negative solution to the analogue of Hilbert's tenth problem for rings of complex entire functions of finite order having lacunary power series expansion at the origin.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"205-209"},"PeriodicalIF":0.4,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570539","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 implicative-infimum subreducts of weak Heyting algebras","authors":"Sergio Celani,&nbsp;Hernán J. San Martín","doi":"10.1002/malq.202300021","DOIUrl":"10.1002/malq.202300021","url":null,"abstract":"<p>The variety of weak Heyting algebras was introduced in 2005 by Celani and Jansana. This corresponds to the strict implication fragment of the normal modal logic <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> which is also known as the subintuitionistic local consequence of the class of all Kripke models. Subresiduated lattices are a generalization of Heyting algebras and particular cases of weak Heyting algebras. They were introduced during the 1970s by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. In this paper we study the class of implicative-infimum subreducts of weak Heyting algebras. In particular, we prove that this class is a variety by giving an equational base for it. We also present a topological duality for the algebraic category whose objects are the implicative-infimum subreducts of subresiduated lattices.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"178-196"},"PeriodicalIF":0.4,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570536","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":"Implications of Ramsey Choice principles in \u0000 \u0000 ZF\u0000 $mathsf {ZF}$","authors":"Lorenz Halbeisen,&nbsp;Riccardo Plati,&nbsp;Saharon Shelah","doi":"10.1002/malq.202300024","DOIUrl":"10.1002/malq.202300024","url":null,"abstract":"&lt;p&gt;The Ramsey Choice principle for families of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-element sets, denoted &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;RC&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$operatorname{RC}_{n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, states that every infinite set &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;annotation&gt;$X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; has an infinite subset &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Ysubseteq X$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with a choice function on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;{&lt;/mo&gt;\u0000 &lt;mi&gt;z&lt;/mi&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mi&gt;z&lt;/mi&gt;\u0000 &lt;mo&gt;|&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;}&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$[Y]^n:= lbrace zsubseteq Y: |z| = nrbrace$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We investigate for which positive integers &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the implication &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;RC&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;⇒&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mo&gt;RC&lt;/mo&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$operatorname{RC}_{m} implies operatorname{RC}_{n}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is provable in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"255-261"},"PeriodicalIF":0.4,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570535","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":"On dense, locally finite subgroups of the automorphism group of certain homogeneous structures","authors":"Gábor Sági","doi":"10.1002/malq.202200060","DOIUrl":"10.1002/malq.202200060","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> be a countable structure such that each finite partial isomorphism of it can be extended to an automorphism. Evans asked if the age (set of finite substructures) of <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> satisfies Hrushovski's extension property, then is it true that the automorphism group <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 </mrow>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{{it Aut}}(mathcal {A})$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> contains a dense, locally finite subgroup? In order to investigate this question, in the previous decades a coherent variant of Hrushovski's extension property has been introduced and studied. Among other results, we provide equivalent conditions for the existence of a dense, locally finite subgroup of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 </mrow>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{{it Aut}}(mathcal {A})$</annotation>\u0000 </semantics></math> in terms of a (new) variant of the coherent extension property. We also compare our notion with other coherent extension properties.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"162-172"},"PeriodicalIF":0.4,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548253","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}