Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set-theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.
{"title":"Discrete subgroups of normed spaces are free","authors":"Tomasz Kania, Ziemowit Kostana","doi":"arxiv-2408.03226","DOIUrl":"https://doi.org/arxiv-2408.03226","url":null,"abstract":"Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that every\u0000countable discrete subgroup of the additive group of a normed space is free\u0000Abelian, hence isomorphic to the direct sum of a certain number of copies of\u0000the additive group of the integers. In the present paper, we take a\u0000set-theoretic approach based on the theory of elementary submodels and the\u0000Singular Compactness Theorem to remove the cardinality constraint from their\u0000result and prove that indeed every discrete subgroup of the additive group of a\u0000normed space is free Abelian.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A group is Artinian if there is no infinite strictly descending chain of subgroups. Ol'shanskii has asked whether there are Artinian groups of arbitrarily large cardinality. We reduce this problem to an analogous question, regarding universal algebras, asked by J'onsson in the 1960s. We provide Artinian groups of cardinality $aleph_n$ for each natural number $n$. We also give a consistent strong negative answer to the question of Ol'shanskii (from a large cardinal assumption) as well as a consistent positive answer. Thus, answers to the questions of Ol'shanskii and J'onsson are independent of set theory.
{"title":"Artinian groups of large cardinality","authors":"Samuel M. Corson, Saharon Shelah","doi":"arxiv-2408.03201","DOIUrl":"https://doi.org/arxiv-2408.03201","url":null,"abstract":"A group is Artinian if there is no infinite strictly descending chain of\u0000subgroups. Ol'shanskii has asked whether there are Artinian groups of\u0000arbitrarily large cardinality. We reduce this problem to an analogous question,\u0000regarding universal algebras, asked by J'onsson in the 1960s. We provide\u0000Artinian groups of cardinality $aleph_n$ for each natural number $n$. We also\u0000give a consistent strong negative answer to the question of Ol'shanskii (from a\u0000large cardinal assumption) as well as a consistent positive answer. Thus,\u0000answers to the questions of Ol'shanskii and J'onsson are independent of set\u0000theory.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we show that forcings which are strongly proper for stationarily many countable elementary submodels preserve each of the following properties 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.
{"title":"Preservation of Topological Properties by Strongly Proper Forcings","authors":"Thomas Gilton, Jared Holshouser","doi":"arxiv-2408.02495","DOIUrl":"https://doi.org/arxiv-2408.02495","url":null,"abstract":"In this paper we show that forcings which are strongly proper for\u0000stationarily many countable elementary submodels preserve each of the following\u0000properties of topological spaces: countably tight; Lindel\"of; Rothberger;\u0000Menger; and a strategic version of Rothberger. This extends results from Dow,\u0000as well as from Iwasa and from Kada.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An ultrafilter is a maximal filter on a set, playing a crucial role in set theory and topology for rigorously handling limits, convergence, and compactness. A connectivity system is defined as a pair (X, f), where X is a finite set and f is a symmetric submodular function. Understanding the duality in these parameters helps to elucidate the relationship between different decompositions and measures of a graph's complexity. In this paper, we delve into ultrafilters on connectivity systems, applying Tukey's Lemma to these systems. Additionally, we explore prefilters, ultra-prefilters, and subbases within the context of connectivity systems. Furthermore, we introduce and investigate 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。
{"title":"Some Property of an Ultrafilter and Graph parameters on Connectivity System","authors":"Takaaki Fujita","doi":"arxiv-2408.02299","DOIUrl":"https://doi.org/arxiv-2408.02299","url":null,"abstract":"An ultrafilter is a maximal filter on a set, playing a crucial role in set\u0000theory and topology for rigorously handling limits, convergence, and\u0000compactness. A connectivity system is defined as a pair (X, f), where X is a\u0000finite set and f is a symmetric submodular function. Understanding the duality\u0000in these parameters helps to elucidate the relationship between different\u0000decompositions and measures of a graph's complexity. In this paper, we delve\u0000into ultrafilters on connectivity systems, applying Tukey's Lemma to these\u0000systems. Additionally, we explore prefilters, ultra-prefilters, and subbases\u0000within the context of connectivity systems. Furthermore, we introduce and\u0000investigate new parameters related to width, length, and depth.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Formal Concept Analysis starts from a very basic data structure comprising objects and their attributes. Sometimes, however, it is beneficial to also define attributes of attributes (so-called meta-attributes). In this paper, we use Triadic Formal Concept Analysis(a triadic approach to Formal Concept Analysis) to develop a framework for this kind of meta-modelling in Formal Concept Analysis, including formal definitions and appropriate visualizations.
{"title":"Meta-Modelling in Formal Concept Analysis","authors":"Yingjian Wang","doi":"arxiv-2408.02435","DOIUrl":"https://doi.org/arxiv-2408.02435","url":null,"abstract":"Formal Concept Analysis starts from a very basic data structure comprising\u0000objects and their attributes. Sometimes, however, it is beneficial to also\u0000define attributes of attributes (so-called meta-attributes). In this paper, we\u0000use Triadic Formal Concept Analysis(a triadic approach to Formal Concept\u0000Analysis) to develop a framework for this kind of meta-modelling in Formal\u0000Concept Analysis, including formal definitions and appropriate visualizations.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stepan L. Kuznetsov, Tikhon Pshenitsyn, Stanislav O. Speranski
The class of all $ast$-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras - ranging from the equational theory to the Horn one, with restricted fragments of the latter in between - was analyzed by Kozen (2002). This paper deals with similar problems for $ast$-continuous residuated Kleene lattices, also called $ast$-continuous action lattices, where the product operation is augmented by adding residuals. We prove that in the presence of residuals the fragment of the corresponding Horn theory with $ast$-free hypotheses has the same complexity as the $omega^omega$ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions 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 as that for $ast$-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and the upper bounds are obtained for the latter ones.
{"title":"Reasoning from hypotheses in *-continuous action lattices","authors":"Stepan L. Kuznetsov, Tikhon Pshenitsyn, Stanislav O. Speranski","doi":"arxiv-2408.02118","DOIUrl":"https://doi.org/arxiv-2408.02118","url":null,"abstract":"The class of all $ast$-continuous Kleene algebras, whose description\u0000includes an infinitary condition on the iteration operator, plays an important\u0000role in computer science. The complexity of reasoning in such algebras -\u0000ranging from the equational theory to the Horn one, with restricted fragments\u0000of the latter in between - was analyzed by Kozen (2002). This paper deals with\u0000similar problems for $ast$-continuous residuated Kleene lattices, also called\u0000$ast$-continuous action lattices, where the product operation is augmented by\u0000adding residuals. We prove that in the presence of residuals the fragment of\u0000the corresponding Horn theory with $ast$-free hypotheses has the same\u0000complexity as the $omega^omega$ iteration of the halting problem, and hence\u0000is properly hyperarithmetical. We also prove that if only commutativity\u0000conditions are allowed as hypotheses, then the complexity drops down to\u0000$Pi^0_1$ (i.e. the complement of the halting problem), which is the same as\u0000that for $ast$-continuous Kleene algebras. In fact, we get stronger upper\u0000bound results: the fragments under consideration are translated into suitable\u0000fragments of infinitary action logic with exponentiation, and the upper bounds\u0000are obtained for the latter ones.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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, it factors, 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 prime equivalence relation, answering a question on Clemens.
{"title":"Generic dichotomy for homomorphisms for $E_0^mathbb{N}$","authors":"Assaf Shani","doi":"arxiv-2408.01261","DOIUrl":"https://doi.org/arxiv-2408.01261","url":null,"abstract":"We prove the following dichotomy. Given an analytic equivalence relation $E$,\u0000either ${E_0^{mathbb{N}}}leq_B{E}$ or else any Borel homomorphism from\u0000$E_0^{mathbb{N}}$ to $E$ is \"very far from a reduction\", specifically, it\u0000factors, on a comeager set, through the projection map\u0000$(2^{mathbb{N}})^{mathbb{N}}to (2^{mathbb{N}})^k$ for some\u0000$kinmathbb{N}$. As a corollary, we prove that $E_0^{mathbb{N}}$ is a prime\u0000equivalence relation, answering a question on Clemens.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Intersection models of generic extensions obtained from a commutative projection systems of notions of forcing has recently regained interest, especially in the study of descriptive set theory. Here, we show that it provides a fruitful framework that opens the door to solving some open problems concerning compactness principles of small cardinals. To exemplify, from suitable assumptions, we construct intersection models satisfying ZFC and any of the following: 1. There is a weakly compact cardinal $kappa$ carrying an indecomposable ultrafilter, yet $kappa$ is not measurable. This answers a question of Ketonen from the late 1970's. 2. For proper class many cardinals $lambda$, the least $lambda$-strongly compact cardinal is singular. This answers a question of Bagaria and Magidor who asked for merely two such cardinals. 3. There is a strongly inaccessible cardinal whose $C$-sequence number is a singular cardinal. This answers a question of Lambie-Hanson and the first author.
{"title":"Ketonen's question and other cardinal sins","authors":"Assaf Rinot, Zhixing You, Jiachen Yuan","doi":"arxiv-2408.01547","DOIUrl":"https://doi.org/arxiv-2408.01547","url":null,"abstract":"Intersection models of generic extensions obtained from a commutative\u0000projection systems of notions of forcing has recently regained interest,\u0000especially in the study of descriptive set theory. Here, we show that it\u0000provides a fruitful framework that opens the door to solving some open problems\u0000concerning compactness principles of small cardinals. To exemplify, from\u0000suitable assumptions, we construct intersection models satisfying ZFC and any\u0000of the following: 1. There is a weakly compact cardinal $kappa$ carrying an indecomposable\u0000ultrafilter, yet $kappa$ is not measurable. This answers a question of Ketonen\u0000from the late 1970's. 2. For proper class many cardinals $lambda$, the least $lambda$-strongly\u0000compact cardinal is singular. This answers a question of Bagaria and Magidor\u0000who asked for merely two such cardinals. 3. There is a strongly inaccessible cardinal whose $C$-sequence number is a\u0000singular cardinal. This answers a question of Lambie-Hanson and the first\u0000author.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fujimoto and Halbach had introduced a novel theory of type-free truth CD which satisfies full classical compositional clauses for connectives and quantifiers. Answering their question, we show that the induction-free variant of 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。
{"title":"Classical determinate truth without induction","authors":"Bartosz Wcisło","doi":"arxiv-2408.01198","DOIUrl":"https://doi.org/arxiv-2408.01198","url":null,"abstract":"Fujimoto and Halbach had introduced a novel theory of type-free truth CD\u0000which satisfies full classical compositional clauses for connectives and\u0000quantifiers. Answering their question, we show that the induction-free variant\u0000of that theory is conservative over Peano Arithmetic.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give improved bounds on the Hausdorff dimension of pinned distance sets of planar sets with dimension strictly less than one. As the planar set becomes more regular (i.e., the Hausdorff and packing dimension become closer), our lower bound on the Hausdorff dimension of the pinned distance set improves. Additionally, we prove the existence of small universal sets 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 also compact 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*}
{"title":"Pinned distances of planar sets with low dimension","authors":"Jacob B. Fiedler, D. M. Stull","doi":"arxiv-2408.00889","DOIUrl":"https://doi.org/arxiv-2408.00889","url":null,"abstract":"In this paper, we give improved bounds on the Hausdorff dimension of pinned\u0000distance sets of planar sets with dimension strictly less than one. As the\u0000planar set becomes more regular (i.e., the Hausdorff and packing dimension\u0000become closer), our lower bound on the Hausdorff dimension of the pinned\u0000distance set improves. Additionally, we prove the existence of small universal\u0000sets for pinned distances. In particular, we show that, if a Borel set\u0000$Xsubseteqmathbb{R}^2$ is weakly regular ($dim_H(X) = dim_P(X)$), and\u0000$dim_H(X) > 1$, 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 also\u0000compact and Alfors-David regular, then for every Borel set\u0000$Ysubseteqmathbb{R}^2$, there exists some $xin X$ such that begin{equation*} dim_H(Delta_x Y) = min{dim_H(Y), 1}. end{equation*}","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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