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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":"62 7-8","pages":"1001 - 1032"},"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":"62 7-8","pages":"991 - 1000"},"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":"62 7-8","pages":"947 - 963"},"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":"62 7-8","pages":"965 - 990"},"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
Definable Tietze extension property in o-minimal expansions of ordered groups 有序群o-极小展开中可定义的Tietze扩张性质
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-15 DOI: 10.1007/s00153-023-00875-5
Masato Fujita
{"title":"Definable Tietze extension property in o-minimal expansions of ordered groups","authors":"Masato Fujita","doi":"10.1007/s00153-023-00875-5","DOIUrl":"10.1007/s00153-023-00875-5","url":null,"abstract":"<div><p>The following two assertions are equivalent for an o-minimal expansion of an ordered group <span>(mathcal M=(M,&lt;,+,0,ldots ))</span>. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function <span>(f:A rightarrow M)</span> defined on a definable closed subset of <span>(M^n)</span> has a definable continuous extension <span>(F:M^n rightarrow M)</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"941 - 945"},"PeriodicalIF":0.3,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44364332","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
Correction to: Towers, mad families, and unboundedness 更正:塔，疯狂的家庭，和无边界
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-04-03 DOI: 10.1007/s00153-023-00870-w
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
{"title":"Correction to: Towers, mad families, and unboundedness","authors":"Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky","doi":"10.1007/s00153-023-00870-w","DOIUrl":"https://doi.org/10.1007/s00153-023-00870-w","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1","pages":"1159-1160"},"PeriodicalIF":0.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52099430","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
Correction to: Towers, mad families, and unboundedness 更正:塔，疯狂的家庭，和无边界
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-04-03 DOI: 10.1007/s00153-023-00870-w
Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky
{"title":"Correction to: Towers, mad families, and unboundedness","authors":"Vera Fischer,&nbsp;Marlene Koelbing,&nbsp;Wolfgang Wohofsky","doi":"10.1007/s00153-023-00870-w","DOIUrl":"10.1007/s00153-023-00870-w","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1159 - 1160"},"PeriodicalIF":0.3,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00870-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50005182","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
Consistency and interpolation in linear continuous logic 线性连续逻辑中的一致性与插值
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-03-21 DOI: 10.1007/s00153-023-00869-3
Mahya Malekghasemi, Seyed-Mohammad Bagheri
{"title":"Consistency and interpolation in linear continuous logic","authors":"Mahya Malekghasemi,&nbsp;Seyed-Mohammad Bagheri","doi":"10.1007/s00153-023-00869-3","DOIUrl":"10.1007/s00153-023-00869-3","url":null,"abstract":"<div><p>We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"931 - 939"},"PeriodicalIF":0.3,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43271849","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
The axiom of choice in metric measure spaces and maximal (delta )-separated sets 度量度量空间和极大(delta )分离集中的选择公理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-03-13 DOI: 10.1007/s00153-023-00868-4
Michał Dybowski, Przemysław Górka
{"title":"The axiom of choice in metric measure spaces and maximal (delta )-separated sets","authors":"Michał Dybowski,&nbsp;Przemysław Górka","doi":"10.1007/s00153-023-00868-4","DOIUrl":"10.1007/s00153-023-00868-4","url":null,"abstract":"<div><p>We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal <span>(delta )</span>-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 5-6","pages":"735 - 749"},"PeriodicalIF":0.3,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00868-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50048305","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
A topological completeness theorem for transfinite provability logic 超限可证明性逻辑的拓扑完备性定理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-02-22 DOI: 10.1007/s00153-023-00863-9
Juan P. Aguilera
{"title":"A topological completeness theorem for transfinite provability logic","authors":"Juan P. Aguilera","doi":"10.1007/s00153-023-00863-9","DOIUrl":"10.1007/s00153-023-00863-9","url":null,"abstract":"<div><p>We prove a topological completeness theorem for the modal logic <span>(textsf{GLP})</span> containing operators <span>({langle xi rangle :xi in textsf{Ord}})</span> intended to capture a wellordered sequence of consistency operators increasing in strength. More specifically, we prove that, given a tall-enough scattered space <i>X</i>, any sentence <span>(phi )</span> consistent with <span>(textsf{GLP})</span> can be satisfied on a polytopological space based on finitely many Icard topologies constructed over <i>X</i> and corresponding to the finitely many modalities that occur in <span>(phi )</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 5-6","pages":"751 - 788"},"PeriodicalIF":0.3,"publicationDate":"2023-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00863-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50042104","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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