The interplay between algebra and logic

代数与逻辑之间的相互作用

• 批准号: 52016048
• 负责人: Professor Dr. Ralf Schindler
• 金额: --
• 依托单位: Fakultät für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/52016048?language=en
• 关键词: interplay; between; algebra; logic

In this project we focus on the interplay between algebra and logic. Over the last decades algebraists have used infinite combinatorics and set-theoretic as well as model-theoretic methods to solve long-standing problems in algebra by showing their consistency with, respectively independence of the usual set theory given by the Zermelo-Fraenkel axioms and the axiom of choice (ZFC). The construction of models with certain combinatorial principles like prediction principles, uniformization principles, etc. by forcing over models in which certain large cardinals exist, played an essential role. Very often algebraic properties are equivalent to set-theoretic statements, for instance cardinal conditions like the (non-)validity of the (generalized) continuum hypothesis. It is the aim of this project to further investigate this interplay between algebra and logic. In particular we shall focus on the structure of infinite rank Butler modules, large cardinals, the automorphism tower problem and dual groups in various models of ZFC obtained by forcing.

在这个项目中，我们关注代数和逻辑之间的相互作用。在过去的几十年里，代数学家使用无限组合学和集合论以及模型论的方法来解决代数中长期存在的问题，他们分别证明了它们与Zermelo-Fraenkel公理和选择公理(ZFC)所给出的通常集合论的一致性。通过强迫存在某些大基数的模型来构建具有某些组合原则的模型，如预测原则、统一原则等，起到了至关重要的作用。通常，代数性质等同于集合论的陈述，例如，像(广义)连续统假设的(非)有效性这样的基本条件。这个项目的目的是进一步研究代数和逻辑之间的相互作用。特别地，我们将重点讨论无限秩巴特勒模的结构、大基数、自同构塔问题和通过强迫得到的各种ZFC模型中的对偶群。

项目成果

Iteratively Changing the Heights of Automorphism Towers

迭代地改变自同构塔的高度

• DOI: 10.1215/00294527-1715662
• 发表时间: 2012
• 期刊: Notre Dame J. Formal Log.
• 作者: G. Fuchs;P. Lucke
• 通讯作者: P. Lucke

ℵn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL*

完全离散估值域上的自由模块，具有几乎微不足道的对偶*

• DOI: 10.1017/s0017089512000614
• 发表时间: 2013
• 期刊: Glasgow Mathematical Journal
• 影响因子: 0.5
• 作者: R. Gobel;J. Rodriguez;L. Strungmann
• 通讯作者: L. Strungmann