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Discrete subgroups of normed spaces are free 规范空间的离散子群是自由的
Pub Date : 2024-08-06 DOI: arxiv-2408.03226
Tomasz Kania, Ziemowit Kostana
Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that everycountable discrete subgroup of the additive group of a normed space is freeAbelian, hence isomorphic to the direct sum of a certain number of copies ofthe additive group of the integers. In the present paper, we take aset-theoretic approach based on the theory of elementary submodels and theSingular Compactness Theorem to remove the cardinality constraint from theirresult and prove that indeed every discrete subgroup of the additive group of anormed space is free Abelian.
Ancel、Dobrowolski 和 Grabowski(Studia Math.,1994)证明了规范空间的加法群的每一个可数离散子群都是自由阿贝尔群,因此与整数加法群的一定数量副本的直和同构。在本文中,我们采用基于基本子模型理论和星形紧凑性定理的集合论方法,消除了他们的结果中的万有引力约束,并证明了规范空间的加法群的每个离散子群确实都是自由阿贝尔群。
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引用次数: 0
Artinian groups of large cardinality 大卡数阿蒂尼群
Pub Date : 2024-08-06 DOI: arxiv-2408.03201
Samuel M. Corson, Saharon Shelah
A group is Artinian if there is no infinite strictly descending chain ofsubgroups. Ol'shanskii has asked whether there are Artinian groups ofarbitrarily large cardinality. We reduce this problem to an analogous question,regarding universal algebras, asked by J'onsson in the 1960s. We provideArtinian groups of cardinality $aleph_n$ for each natural number $n$. We alsogive a consistent strong negative answer to the question of Ol'shanskii (from alarge cardinal assumption) as well as a consistent positive answer. Thus,answers to the questions of Ol'shanskii and J'onsson are independent of settheory.
如果不存在无限严格降序的子群链,那么这个群就是阿蒂尼亚群。奥尔尚斯基曾提出是否存在任意大心数的阿蒂尼亚群。我们把这个问题还原为一个类似的问题,即 J'onsson 在 20 世纪 60 年代提出的关于普遍代数的问题。我们为每个自然数 $n$ 提供了心数为 $aleph_n$ 的阿蒂尼亚群。我们还对奥尔尚斯基的问题给出了一致的强否定答案(来自大心性假设)和一致的肯定答案。因此,对奥尔尚斯基和约翰逊问题的回答是独立于设定理论的。
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引用次数: 0
Preservation of Topological Properties by Strongly Proper Forcings 强适当强制力对拓扑特性的保护
Pub Date : 2024-08-05 DOI: arxiv-2408.02495
Thomas Gilton, Jared Holshouser
In this paper we show that forcings which are strongly proper forstationarily many countable elementary submodels preserve each of the followingproperties of topological spaces: countably tight; Lindel"of; Rothberger;Menger; and a strategic version of Rothberger. This extends results from Dow,as well as from Iwasa and from Kada.
在本文中,我们证明了对于有一定数量的可数基本子模型来说是强适当的强制力保留了拓扑空间的以下每个性质:可数紧密性;林德尔的;罗斯伯格的;门格尔的;以及罗斯伯格的策略版本。这扩展了道以及岩佐和加田的结果。
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引用次数: 0
Some Property of an Ultrafilter and Graph parameters on Connectivity System 超滤器的一些特性和连接系统上的图形参数
Pub Date : 2024-08-05 DOI: arxiv-2408.02299
Takaaki Fujita
An ultrafilter is a maximal filter on a set, playing a crucial role in settheory and topology for rigorously handling limits, convergence, andcompactness. A connectivity system is defined as a pair (X, f), where X is afinite set and f is a symmetric submodular function. Understanding the dualityin these parameters helps to elucidate the relationship between differentdecompositions and measures of a graph's complexity. In this paper, we delveinto ultrafilters on connectivity systems, applying Tukey's Lemma to thesesystems. Additionally, we explore prefilters, ultra-prefilters, and subbaseswithin the context of connectivity systems. Furthermore, we introduce andinvestigate new parameters related to width, length, and depth.
超滤波器是集合上的最大滤波器,在集合论和拓扑学中对严格处理极限、收敛和紧凑性起着至关重要的作用。连通性系统被定义为一对(X, f),其中 X 是一个无穷集,f 是一个对称子模函数。理解这些参数的对偶性有助于阐明不同分解与图的复杂性度量之间的关系。在本文中,我们深入探讨了连通性系统上的超滤波器,并将 Tukey's Lemma 应用于这些系统。此外,我们还探讨了连通性系统中的前置过滤器、超前置过滤器和子基础。此外,我们还引入并研究了与宽度、长度和深度相关的新参数。
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引用次数: 0
Meta-Modelling in Formal Concept Analysis 形式概念分析中的元建模
Pub Date : 2024-08-05 DOI: arxiv-2408.02435
Yingjian Wang
Formal Concept Analysis starts from a very basic data structure comprisingobjects and their attributes. Sometimes, however, it is beneficial to alsodefine attributes of attributes (so-called meta-attributes). In this paper, weuse Triadic Formal Concept Analysis(a triadic approach to Formal ConceptAnalysis) to develop a framework for this kind of meta-modelling in FormalConcept Analysis, including formal definitions and appropriate visualizations.
正式概念分析从一个由对象及其属性组成的非常基本的数据结构开始。然而,有时定义属性的属性(即所谓的元属性)也是有益的。在本文中,我们利用三元形式概念分析(形式概念分析的一种三元方法)为形式概念分析中的这种元建模开发了一个框架,包括形式定义和适当的可视化。
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引用次数: 0
Reasoning from hypotheses in *-continuous action lattices 从*连续作用网格中的假设推理
Pub Date : 2024-08-04 DOI: arxiv-2408.02118
Stepan L. Kuznetsov, Tikhon Pshenitsyn, Stanislav O. Speranski
The class of all $ast$-continuous Kleene algebras, whose descriptionincludes an infinitary condition on the iteration operator, plays an importantrole in computer science. The complexity of reasoning in such algebras -ranging from the equational theory to the Horn one, with restricted fragmentsof the latter in between - was analyzed by Kozen (2002). This paper deals withsimilar problems for $ast$-continuous residuated Kleene lattices, also called$ast$-continuous action lattices, where the product operation is augmented byadding residuals. We prove that in the presence of residuals the fragment ofthe corresponding Horn theory with $ast$-free hypotheses has the samecomplexity as the $omega^omega$ iteration of the halting problem, and henceis properly hyperarithmetical. We also prove that if only commutativityconditions are allowed as hypotheses, then the complexity drops down to$Pi^0_1$ (i.e. the complement of the halting problem), which is the same asthat for $ast$-continuous Kleene algebras. In fact, we get stronger upperbound results: the fragments under consideration are translated into suitablefragments of infinitary action logic with exponentiation, and the upper boundsare obtained for the latter ones.
所有$ast$-连续克莱因布拉的描述都包括迭代算子的无穷条件,这类布拉在计算机科学中发挥着重要作用。Kozen (2002)分析了在这类代数中推理的复杂性--从等式理论到霍恩理论,中间还有后者的有限片段。本文讨论了 $ast$-continuous residuated Kleene lattices(也称为 $ast$-continuous action lattices)的类似问题,其中乘积运算是通过添加残差来增强的。我们证明,在有残差的情况下,相应的无$ast$假设的霍恩理论片段具有与停止问题的$omega^omega$迭代相同的复杂性,因此是适当的超算术的。我们还证明,如果只允许交换性条件作为假设,那么复杂度会下降到$/Pi^0_1$(即停止问题的补码),这与$/ast$-连续克莱因代数的复杂度相同。事实上,我们得到了更强的上界结果:所考虑的片段被转化为带指数化的无穷行动逻辑的合适片段,而上界是针对后一种片段得到的。
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引用次数: 0
Generic dichotomy for homomorphisms for $E_0^mathbb{N}$ 同态$E_0^mathbb{N}$的通用二分法
Pub Date : 2024-08-02 DOI: arxiv-2408.01261
Assaf Shani
We prove the following dichotomy. Given an analytic equivalence relation $E$,either ${E_0^{mathbb{N}}}leq_B{E}$ or else any Borel homomorphism from$E_0^{mathbb{N}}$ to $E$ is "very far from a reduction", specifically, itfactors, on a comeager set, through the projection map$(2^{mathbb{N}})^{mathbb{N}}to (2^{mathbb{N}})^k$ for some$kinmathbb{N}$. As a corollary, we prove that $E_0^{mathbb{N}}$ is a primeequivalence relation, answering a question on Clemens.
我们将证明以下二分法。给定一个解析等价关系 $E$,要么 ${E_0^{mathbb{N}}}leq_B{E}$ 要么任何从$E_0^{mathbb{N}}$ 到 $E$ 的伯尔同构都 "离还原很远"、具体地说,它是通过投影图$(2^{mathbb{N}})^{mathbb{N}}to (2^{mathbb{N}})^k$ 对某个$kinmathbb{N}$的投影图,在一个comeager集合上形成的。作为一个推论,我们证明 $E_0^{mathbb{N}}$ 是一个首要等价关系,从而回答了一个关于克莱门斯的问题。
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引用次数: 0
Ketonen's question and other cardinal sins 凯托宁的问题和其他大罪
Pub Date : 2024-08-02 DOI: arxiv-2408.01547
Assaf Rinot, Zhixing You, Jiachen Yuan
Intersection models of generic extensions obtained from a commutativeprojection systems of notions of forcing has recently regained interest,especially in the study of descriptive set theory. Here, we show that itprovides a fruitful framework that opens the door to solving some open problemsconcerning compactness principles of small cardinals. To exemplify, fromsuitable assumptions, we construct intersection models satisfying ZFC and anyof the following: 1. There is a weakly compact cardinal $kappa$ carrying an indecomposableultrafilter, yet $kappa$ is not measurable. This answers a question of Ketonenfrom the late 1970's. 2. For proper class many cardinals $lambda$, the least $lambda$-stronglycompact cardinal is singular. This answers a question of Bagaria and Magidorwho asked for merely two such cardinals. 3. There is a strongly inaccessible cardinal whose $C$-sequence number is asingular cardinal. This answers a question of Lambie-Hanson and the firstauthor.
从强制概念的交换投影系统得到的泛函扩展的交集模型最近重新引起了人们的兴趣,特别是在描述集合论的研究中。在这里,我们证明它提供了一个富有成果的框架,为解决一些有关小红心的紧凑性原则的未决问题打开了大门。举例来说,从合适的假设出发,我们构建了满足 ZFC 和以下任意条件的交集模型:1.有一个弱紧凑的红心$kappa$携带一个不可分解的超滤波器,然而$kappa$是不可测的。这回答了克托宁在 20 世纪 70 年代末提出的一个问题。2.对于适当的多类红心$lambda$,最小的$lambda$-强紧密红心是奇异的。这回答了巴加利亚和马吉多尔的一个问题,他们只要求两个这样的红心。3.有一个强不可及的红心,它的$C$序列数是奇异红心。这回答了兰比-汉森和第一作者的一个问题。
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引用次数: 0
Classical determinate truth without induction 无需归纳的经典确定性真理
Pub Date : 2024-08-02 DOI: arxiv-2408.01198
Bartosz Wcisło
Fujimoto and Halbach had introduced a novel theory of type-free truth CDwhich satisfies full classical compositional clauses for connectives andquantifiers. Answering their question, we show that the induction-free variantof that theory is conservative over Peano Arithmetic.
藤本和哈尔巴赫提出了一种新颖的无类型真值 CD 理论,它满足连接词和量词的全经典组成子句。为了回答他们的问题,我们证明了该理论的无归纳变体在皮亚诺算术上是保守的。
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引用次数: 0
Pinned distances of planar sets with low dimension 低维度平面集的钉距
Pub Date : 2024-08-01 DOI: arxiv-2408.00889
Jacob B. Fiedler, D. M. Stull
In this paper, we give improved bounds on the Hausdorff dimension of pinneddistance sets of planar sets with dimension strictly less than one. As theplanar set becomes more regular (i.e., the Hausdorff and packing dimensionbecome closer), our lower bound on the Hausdorff dimension of the pinneddistance set improves. Additionally, we prove the existence of small universalsets for pinned distances. In particular, we show that, if a Borel set$Xsubseteqmathbb{R}^2$ is weakly regular ($dim_H(X) = dim_P(X)$), and$dim_H(X) > 1$, then begin{equation*} suplimits_{xin X}dim_H(Delta_x Y) = min{dim_H(Y), 1} end{equation*} for every Borel set $Ysubseteqmathbb{R}^2$. Furthermore, if $X$ is alsocompact and Alfors-David regular, then for every Borel set$Ysubseteqmathbb{R}^2$, there exists some $xin X$ such that begin{equation*} dim_H(Delta_x Y) = min{dim_H(Y), 1}. end{equation*}
在本文中,我们给出了维度严格小于 1 的平面集的钉距集的豪斯多夫维度的改进边界。随着平面集变得越来越规则(即 Hausdorff 维度和堆积维度越来越接近),我们对钉扎距离集的 Hausdorff 维度的下界也随之提高。此外,我们还证明了钉住距离小普遍集的存在。我们特别证明了,如果一个波尔集合$X(subseteq/mathbb{R}^2$是弱正则的($dim_H(X) = dim_P(X)$),并且$dim_H(X) > 1$,那么}suplimits_{xin X}dim_H(Delta_x Y) = min{dim_H(Y), 1}end{equation*} for every Borel set $Ysubseteqmathbb{R}^2$.此外,如果 $X$ 也是紧凑的、Alfors-David 正则的,那么对于每个 Borel 集$Y(subseteqmathbb{R}^2$),在 X$ 中存在一些 $xin ,使得 begin{equation*}dim_H(Delta_x Y) = min{dim_H(Y), 1}.end{equation*}
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arXiv - MATH - Logic
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