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The fixed point and the Craig interpolation properties for sublogics of $$textbf{IL}$$ $$textbf{IL}子逻辑的不动点和Craig插值性质$$
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-10 DOI: 10.1007/s00153-023-00882-6
Sohei Iwata, Taishi Kurahashi, Yuya Okawa
{"title":"The fixed point and the Craig interpolation properties for sublogics of $$textbf{IL}$$","authors":"Sohei Iwata, Taishi Kurahashi, Yuya Okawa","doi":"10.1007/s00153-023-00882-6","DOIUrl":"https://doi.org/10.1007/s00153-023-00882-6","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43405479","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: 1
On the complexity of the theory of a computably presented metric structure 论可计算度量结构理论的复杂性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-09 DOI: 10.1007/s00153-023-00884-4
Caleb Camrud, Isaac Goldbring, Timothy H. McNicholl
{"title":"On the complexity of the theory of a computably presented metric structure","authors":"Caleb Camrud,&nbsp;Isaac Goldbring,&nbsp;Timothy H. McNicholl","doi":"10.1007/s00153-023-00884-4","DOIUrl":"10.1007/s00153-023-00884-4","url":null,"abstract":"<div><p>We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the <i>closed</i> diagram, which encapsulates weak inequalities of the form <span>(phi ^mathcal {M}le r)</span>, and the <i>open</i> diagram, which encapsulates strict inequalities of the form <span>(phi ^mathcal {M}&lt; r)</span>. We show that the closed and open <span>(Sigma _N)</span> diagrams are <span>(Pi ^0_{N+1})</span> and <span>(Sigma ^0_N)</span> respectively, and that the closed and open <span>(Pi _N)</span> diagrams are <span>(Pi ^0_N)</span> and <span>(Sigma ^0_{N + 1})</span> respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47542236","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: 2
Recursive Polish spaces 递归抛光空间
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-09 DOI: 10.1007/s00153-023-00883-5
Tyler Arant
{"title":"Recursive Polish spaces","authors":"Tyler Arant","doi":"10.1007/s00153-023-00883-5","DOIUrl":"10.1007/s00153-023-00883-5","url":null,"abstract":"<div><p>This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space <span>({mathcal {X}})</span>, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space <span>(mathbb {N}times {mathcal {X}})</span>.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00883-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795318","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
Structure of semisimple rings in reverse and computable mathematics 半单环的结构在逆向和可计算数学中的应用
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-08 DOI: 10.1007/s00153-023-00885-3
Huishan Wu
{"title":"Structure of semisimple rings in reverse and computable mathematics","authors":"Huishan Wu","doi":"10.1007/s00153-023-00885-3","DOIUrl":"10.1007/s00153-023-00885-3","url":null,"abstract":"<div><p>This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in <span>(mathrm RCA_{0})</span>, showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in <span>(mathrm RCA_{0})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45835876","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
A syntactic approach to Borel functions: some extensions of Louveau’s theorem 博雷尔函数的句法方法:卢沃定理的一些扩展
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-02 DOI: 10.1007/s00153-023-00880-8
Takayuki Kihara, Kenta Sasaki
{"title":"A syntactic approach to Borel functions: some extensions of Louveau’s theorem","authors":"Takayuki Kihara,&nbsp;Kenta Sasaki","doi":"10.1007/s00153-023-00880-8","DOIUrl":"10.1007/s00153-023-00880-8","url":null,"abstract":"<div><p>Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class <span>(Gamma )</span>, then its <span>(Gamma )</span>-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a <span>( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)</span>-function, then one can find its <span>( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)</span>-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00880-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42852550","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
On the non-existence of (kappa )-mad families 关于不存在(kappa )疯狂家庭
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-23 DOI: 10.1007/s00153-023-00874-6
Haim Horowitz, Saharon Shelah
{"title":"On the non-existence of (kappa )-mad families","authors":"Haim Horowitz,&nbsp;Saharon Shelah","doi":"10.1007/s00153-023-00874-6","DOIUrl":"10.1007/s00153-023-00874-6","url":null,"abstract":"<div><p>Starting from a model with a Laver-indestructible supercompact cardinal <span>(kappa )</span>, we construct a model of <span>(ZF+DC_{kappa })</span> where there are no <span>(kappa )</span>-mad families.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00874-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50044367","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
An AEC framework for fields with commuting automorphisms 具有交换自同构域的AEC框架
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-20 DOI: 10.1007/s00153-023-00879-1
Tapani Hyttinen, Kaisa Kangas
{"title":"An AEC framework for fields with commuting automorphisms","authors":"Tapani Hyttinen,&nbsp;Kaisa Kangas","doi":"10.1007/s00153-023-00879-1","DOIUrl":"10.1007/s00153-023-00879-1","url":null,"abstract":"<div><p>In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00879-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45633905","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: 2
The small index property for countable superatomic boolean algebras 可数超原子布尔代数的小指标性质
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-19 DOI: 10.1007/s00153-023-00876-4
J. K. Truss
{"title":"The small index property for countable superatomic boolean algebras","authors":"J. K. Truss","doi":"10.1007/s00153-023-00876-4","DOIUrl":"10.1007/s00153-023-00876-4","url":null,"abstract":"<div><p>It is shown that all the countable superatomic boolean algebras of finite rank have the small index property.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00876-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45156289","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
Questions on cardinal invariants of Boolean algebras 布尔代数的基数不变量问题
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-18 DOI: 10.1007/s00153-023-00872-8
Mario Jardón Santos
{"title":"Questions on cardinal invariants of Boolean algebras","authors":"Mario Jardón Santos","doi":"10.1007/s00153-023-00872-8","DOIUrl":"10.1007/s00153-023-00872-8","url":null,"abstract":"<div><p>In the book Cardinal Invariants on Boolean Algebras by J. Donald Monk many such cardinal functions are defined and studied. Among them several are generalizations of well known cardinal characteristics of the continuum. Alongside a long list of open problems is given. Focusing on half a dozen of those cardinal invariants some of those problems are given an answer here, which in most of the cases is a definitive one. Most of them can be divided in two groups. The problems of the first group ask about the change on those cardinal functions when going from a given infinite Boolean algebra to its simple extensions, while in the second group the comparison is between a couple of given infinite Boolean algebras and their free product.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00872-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44557674","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: 2
Mathias and silver forcing parametrized by density Mathias和silver力的密度参数化
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-18 DOI: 10.1007/s00153-023-00881-7
Giorgio Laguzzi, Heike Mildenberger, Brendan Stuber-Rousselle
{"title":"Mathias and silver forcing parametrized by density","authors":"Giorgio Laguzzi,&nbsp;Heike Mildenberger,&nbsp;Brendan Stuber-Rousselle","doi":"10.1007/s00153-023-00881-7","DOIUrl":"10.1007/s00153-023-00881-7","url":null,"abstract":"<div><p>We define and investigate versions of Silver and Mathias forcing with respect to lower and upper density. We focus on properness, Axiom A, chain conditions, preservation of cardinals and adding Cohen reals. We find rough forcings that collapse <span>(2^omega )</span> to <span>(omega )</span>, while others are surprisingly gentle. We also study connections between regularity properties induced by these parametrized forcing notions and the Baire property.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00881-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44594215","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: 1
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