Mathematical Logic Quarterly最新文献

英文中文

The set of injections and the set of surjections on a set 注入集合和射出集

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-09-16 DOI: 10.1002/malq.202300059

Natthajak Kamkru, Nattapon Sonpanow

In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <math> <semantics> <mi>m</mi> <annotation>$mathfrak {m}$</annotation> </semantics></math>, denoted by <math> <semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <annotation>$I(mathfrak {m})$</annotation> </semantics></math>, <math> <semantics> <mrow> <mi>J</mi> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <annotation>$J(mathfrak {m})$</annotation> </semantics></math> and <math> <semantics> <msup> <mi>m</mi> <mi>m</mi> </msup> <annotation>$mathfrak {m}^mathfrak {m}$</annotation> </semantics></math>, respectively. Among our results, we show that “<math> <semantics> <mrow> <msup> <mo>seq</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mi>I</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mo>seq</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> <annotation>$operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$</annotation> </semantics></math>”, “<math> <semantics> <mrow> <msup> <mo>seq</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mi>J</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</

在本文中，我们研究了注入集合、射出集合和一个集合上所有函数的集合的心数之间的关系，这些集合的心数为 m $mathfrak {m}$，分别用 I ( m ) $I(mathfrak {m})$、J ( m ) $J(mathfrak {m})$和 m m $mathfrak {m}^mathfrak {m}$表示。Among our results, we show that “ seq 1 − 1 ( m ) ≠ I ( m ) ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$ ”, “ seq 1 − 1 ( m ) ≠ J ( m ) ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})ne J(mathfrak {m})ne operatorname{seq}(mathfrak {m})$ ” and “ seq 1 − 1 ( m ) < m m ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})<mathfrak {m}^mathfrak {m}ne operatorname{seq}(mathfrak {m})$ ” are provable for an arbitrary infinite cardinal m $mathfrak {m}$ , and these are the best possible results, in the Zermelo-Fraenkel set theory ( ZF $mathsf {ZF}$ ) without the Axiom of Choice.另外，我们还证明，存在一个无限红心 m $mathfrak {m}$，使得 S ( m ) = I ( m ) <； J ( m ) $S(mathfrak {m})=I(mathfrak {m})<J(mathfrak {m})$ 其中 S ( m ) $S(mathfrak {m})$表示一个集合上双射集合的万有性，这个集合的万有性为 m $mathfrak {m}$ 。

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引用次数: 0

Effectiveness of Walker's cancellation theorem 沃克取消定理的有效性

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-09-13 DOI: 10.1002/malq.202400030

Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie

Walker's cancellation theorem for abelian groups tells us that if <math> <semantics> <mi>A</mi> <annotation>$A$</annotation> </semantics></math> is finitely generated and <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> and <math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math> are such that <math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>G</mi> <mo>≅</mo> <mi>A</mi> <mi>⊕</mi> <mi>H</mi> </mrow> <annotation>$A oplus G cong A oplus H$</annotation> </semantics></math>, then <math> <semantics> <mrow> <mi>G</mi> <mo>≅</mo> <mi>H</mi> </mrow> <annotation>$G cong H$</annotation> </semantics></math>. Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> and <math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math>, given indices for <math> <semantics> <mi>A</mi> <annotation>$A$</annotation> </semantics></math>, <math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math>, <math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math>, the isomorphism between <math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>G</mi> </mrow> <annotation>$A oplus G$</annotation> </semantics></math> and <math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>H</mi> </mrow> <annotation>$A oplus H$</annotation> </semantics></math>, and the rank of <math> <semantics>

沃克的无边际群取消定理告诉我们，如果是有限生成的，且，那么。德沃（Deveau）指出，该定理可以被有效化，但不是均匀地有效化。在本文中，我们对 Deveau 的初步分析进行了扩展，证明在给定、、、之间同构的指数以及、的秩的情况下，统一输出、与之间同构的指数的复杂度为。此外，我们还发现，即使指定了和的副本中的生成器，复杂度依然存在。

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引用次数: 0

Good points for scales (and more) 天平的优点（以及更多）

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-09-09 DOI: 10.1002/malq.202300034

Pierre Matet

Given a scale (in the sense of Shelah's pcf theory), we list various conditions ensuring that a given point is good for the scale.

给定一个标度（根据谢拉的 pcf 理论），我们列出各种条件，确保给定的点对标度有利。

引用次数: 0

Editorial correction for L. Halbeisen, R. Plati, and Saharon Shelah, “Implications of Ramsey Choice principles in ZF $mathsf {ZF}$ ”, https://doi.org/10.1002/malq.202300024 对 L. Halbeisen、R. Plati 和 Saharon Shelah "拉姆齐选择原则在 ZF$mathsf {ZF}$ 中的影响 "的编辑更正，https://doi.org/10.1002/malq.202300024。

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-09-09 DOI: 10.1002/malq.202430002

The numbers of corollaries and propositions in the proof of Theorem 3.8 on p. 260 in the article Implications of Ramsey Choice principles in $� � ZF � �$ by Lorenz Halbeisen, Riccardo Plati, and Saharon Shelah (doi: 10.1002/malq.202300024) are incorrect. The correct numbers are given here:

August 2024

The MLQ Editorial Office

Lorenz Halbeisen、Riccardo Plati 和 Saharon Shelah 所著文章《拉姆齐选择原则在 ZF $mathsf {ZF}$ 中的含义》(doi: 10.1002/malq.202300024)第 260 页中定理 3.8 的证明中的推论和命题编号有误。正确数字如下：2024 年 8 月MLQ编辑部

引用次数: 0

Wadge degrees of Δ 2 0 $mathbf{Delta }^0_2$ omega-powers Δ20$mathbf{Delta }^0_2$ Ω-幂的瓦奇度

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-09-06 DOI: 10.1002/malq.202400024

Olivier Finkel, Dominique Lecomte

We provide, for each natural number <math> <semantics> <mi>n</mi> <annotation>$n$</annotation> </semantics></math> and each class among <math> <semantics> <mrow> <msub> <mi>D</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>Σ</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <annotation>$D_n(mathbf {Sigma }^0_1)$</annotation> </semantics></math>, <math> <semantics> <mrow> <msub> <mover> <mi>D</mi> <mo>̌</mo> </mover> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>Σ</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <annotation>$check{D}_n(mathbf {Sigma }^0_1)$</annotation> </semantics></math>, <math> <semantics> <mrow> <msub> <mi>D</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>Σ</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> <mi>⊕</mi> <msub> <mover> <mi>D</mi> <mo>̌</mo> </mover> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mi>Σ</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mo>)</mo> </mrow> </mrow> <annotation>$D_{2n+1}(mathbf {Sigma }^0_1)oplus check{D}_{2n+1}(mathbf {Sigma }^0_1)$</annotation> </semantics></math>, a regular language whose ass

我们为每个自然数和ⅣⅤ类中的每个类提供一种正则表达式语言，其相关的Ω-幂对该类来说是完整的。

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引用次数: 0

Contents: (Math. Log. Quart. 2/2024) 内容：(数学逻辑学季刊》第 2/2024 期）

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-07-23 DOI: 10.1002/malq.202470022

引用次数: 0

Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups 邻最小结构和半代数群中可定义局部同态的扩展

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-07-17 DOI: 10.1002/malq.202300028

Eliana Barriga

We state conditions for which a definable local homomorphism between two locally definable groups $� � G � �$ , $� � �^{G �' �} � �$ can be uniquely extended when $� � G � �$ 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 $� � G � �$ not necessarily abelian over a sufficiently saturated real closed field $� � R � �$ ; namely, that the o-minimal universal covering group $� � \overset{G}{�} � �$ of $� � G � �$ is an open locally definable subgroup of $� � \overset{� H � �^{� (� R �) � � 0 �} �}{�} � �$ for some $� � R � �$ -algebraic group $� � H � �$ (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group $� � G �$

我们说明了两个局部可定义群 , 之间的可定义局部同态在简单相连时可以唯一扩展的条件（定理 2.1）。作为这一结果的应用，我们得到了 [3, 定理 9.1] 的简便证明（参见推论 2.3）。我们还证明了 [3，定理 10.2] 对于在充分饱和实闭域上的任何可定连通可定紧密半代数群（不一定是无性的）也是成立的；即对于某个-代数群，它的 o-minimal 通用覆盖群是它的一个开放局部可定子群（定理 3.3）。最后，对于一个在上的无性定义相连半代数群，我们将其描述为交换-代数群的 o-minimal 普遍覆盖群的一个局部可定义的扩展子群（定理 3.4）。

{"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":"We state conditions for which a definable local homomorphism between two locally definable groups <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$mathcal {G}$</annotation>\u0000 </semantics></math>, <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 <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 <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> not necessarily abelian over a sufficiently saturated real closed field <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>; namely, that the o-minimal universal covering group <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>G</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <annotation>$widetilde{G}$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is an open locally definable subgroup of <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 <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-algebraic group <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 <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}

引用次数: 0

The Hartogs–Lindenbaum spectrum of symmetric extensions 对称扩展的哈托格-林登鲍姆谱

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-07-16 DOI: 10.1002/malq.202300047

Calliope Ryan-Smith

We expand the classic result that

� � �_{AC � WO �} � �

is equivalent to the statement “For all

� � X � �

� � � ℵ � � (� X �) � � = � �^{ℵ � * �} � � (� X �) � � � �

” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of

� � ZF � �

, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of

� � ZFC � �

. We prove that all such spectra fall into a very rigid pattern.

我们通过证明更多相关陈述的等价性，扩展了等价于 "对于所有Ⅳ"陈述的经典结果。然后，我们引入了Ⅳ模型的哈托格斯-林登鲍姆谱，并考察了通过Ⅳ模型的对称扩展得到的模型中这些谱的结构。我们证明，所有这些谱都属于一种非常严格的模式。

{"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":"We expand the classic result that <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 <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>, <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 <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 <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.","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}

引用次数: 0

Filter-Menger set of reals in Cohen extensions 科恩扩展中的滤门格尔实数集

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-07-15 DOI: 10.1002/malq.202300008

Hang Zhang, Shuguo Zhang

We prove that for every ultrafilter <math> <semantics> <mi>U</mi> <annotation>$mathcal {U}$</annotation> </semantics></math> on <math> <semantics> <mi>ω</mi> <annotation>$omega$</annotation> </semantics></math> there exists a filter <math> <semantics> <mi>F</mi> <annotation>$mathcal {F}$</annotation> </semantics></math> on <math> <semantics> <msup> <mn>2</mn> <mrow> <mo><</mo> <mi>ω</mi> </mrow> </msup> <annotation>$2^{<omega }$</annotation> </semantics></math> which is <math> <semantics> <mi>U</mi> <annotation>$mathcal {U}$</annotation> </semantics></math>-Menger and <math> <semantics> <mrow> <mi>χ</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> <mo>=</mo> <mi>b</mi> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <annotation>$chi (mathcal {F})=mathfrak {b}(mathcal {U})$</annotation> </semantics></math>. We show that in the Cohen model there exists such <math> <semantics> <mi>F</mi> <annotation>$mathcal {F}$</annotation> </semantics></math> 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 <math> <semantics> <mi>d</mi> <annotation>$mathfrak {d}$</annotation> </semantics></math> that is not Hurewicz in the <math> <semantics> <mi>κ</mi> <annotation>$kappa$</annotation> </semantics></math>-Cohen model where <math> <semantics> <mrow> <mi>κ</mi> <mo>></mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> </mrow> <annotation>$kappa >omega _{1}$</annotation> </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <math> <semantics> <mrow> <mi>b</mi> <mo><</mo> <mi>d</mi> </mrow> <annotation>$mathfrak {b}<mathfrak {d}$</annotation> </se

我们证明，对于上的每一个超滤波器，都存在一个滤波器，它是-门格尔和。我们用 Nyikos [10] 的构造证明，在科恩模型中存在这样的高滤波器。这回答了达斯的一个问题[2, 问题 7]。我们证明，在-科恩模型中，存在一个不可数正则表达式的门格尔滤波器的特征不是胡勒维茨。这表明对埃尔南德斯-古铁雷斯和塞普蒂奇[3, 问题 2.8]问题的肯定回答与 .我们还研究了-科恩模型中互为科恩有数集所产生的滤波器。我们证明了地面模型中的和以及每个主族在广延上都是无界的。我们提出了两个问题。

{"title":"Filter-Menger set of reals in Cohen extensions","authors":"Hang Zhang, Shuguo Zhang","doi":"10.1002/malq.202300008","DOIUrl":"10.1002/malq.202300008","url":null,"abstract":"We prove that for every ultrafilter <math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$mathcal {U}$</annotation>\u0000 </semantics></math> on <math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> there exists a filter <math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> on <math>\u0000 <semantics>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mrow>\u0000 <mo><</mo>\u0000 <mi>ω</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$2^{&lt;omega }$</annotation>\u0000 </semantics></math> which is <math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$mathcal {U}$</annotation>\u0000 </semantics></math>-Menger and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mo>(</mo>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>b</mi>\u0000 <mo>(</mo>\u0000 <mi>U</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$chi (mathcal {F})=mathfrak {b}(mathcal {U})$</annotation>\u0000 </semantics></math>. We show that in the Cohen model there exists such <math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> 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 <math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$mathfrak {d}$</annotation>\u0000 </semantics></math> that is not Hurewicz in the <math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-Cohen model where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>></mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$kappa &gt;omega _{1}$</annotation>\u0000 </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 <mo><</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$mathfrak {b}&lt;mathfrak {d}$</annotation>\u0000 </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}

引用次数: 0

Expansions of real closed fields with the Banach fixed point property 具有巴拿赫定点特性的实闭域展开

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly

Pub Date : 2024-07-15 DOI: 10.1002/malq.202400001

Athipat Thamrongthanyalak

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 $� � E � �$ , if every definable contraction on $� � E � �$ has a fixed point, then $� � E � �$ is closed. Let $� � R � �$ be an expansion of a real closed field. We prove that if $� � R � �$ has an o-minimal open core, then it has the Banach fixed point property; and if $� � R � �$ is definably complete and has the Banach fixed point property, then it has a locally o-minimal open core.

我们研究巴拿赫定点定理会话的一个变体及其与实闭域展开中的驯服性的联系。对于每个局部封闭的可定义集合 , 如果其上的每个可定义收缩都有一个定点，则称实闭有序域的展开具有巴拿赫定点性质。设是一个实封闭域的展开式。我们证明，如果有一个 o-minimal 开核，那么它具有巴拿赫定点性质；如果是可定义完全且具有巴拿赫定点性质，那么它有一个局部 o-minimal 开核。

{"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":"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 <math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math>, if every definable contraction on <math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> has a fixed point, then <math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> is closed. Let <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 <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 <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.","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}

引用次数: 0

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