Independence Relations in Abstract Elementary Categories

M. Kamsma
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引用次数: 3

Abstract

Abstract In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation. Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$ , each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique. All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work. We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness. Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing. Abstract prepared by Mark Kamsma. E-mail: mark@markkamsma.nl. URL: https://markkamsma.nl/phd-thesis.
抽象基本范畴中的独立关系
在数学逻辑的一个分支——模型论中,我们可以根据数学结构的逻辑复杂度对其进行分类。这就产生了所谓的稳定性层次结构。独立关系在这种稳定性层次中起着重要作用。一个独立关系告诉我们一个结构的哪些子集包含了彼此的信息,例如,向量空间中的线性独立产生了这样一个关系。稳定性层次结构中一些重要的类是稳定的、简单的和NSOP $_1$,每个类都包含在下一个类中。对于每一类都存在一个所谓的金-皮莱式定理。该定理描述了独立性关系与稳定性层次之间的相互作用。例如,简单性等于承认某种独立关系,而这种独立关系必须是唯一的。所有这些都是典型的一阶逻辑。它的一部分已经被推广到其他框架中,比如连续逻辑,正逻辑,甚至是一个非常普遍的范畴论框架。在本文中,我们继续这项工作。我们介绍了aecat的框架,它是一种特殊的可访问类别。我们证明了在一个AECat中最多只能存在一个稳定的、简单的或NSOP $_1$ -like的独立关系。因此,我们恢复(部分)原始的稳定性层次结构。为此,我们介绍了长分法、isii分法和长金分法的概念,它们是基于经典的分法和金分法的概念,但它们在没有紧凑性的情况下也能很好地工作。转换框架,我们将NSOP $_1$理论中的Kim-dividing推广到正逻辑。我们证明了存在闭模型上的kim - divided具有在全一阶逻辑中已知的所有好的性质。我们还提供了一个完整的Kim-Pillay风格定理:一个正理论是NSOP $_1$当且仅当有一个足够好的独立关系,这个独立关系必须由Kim-dividing给出。摘要由Mark Kamsma准备。电子邮件:mark@markkamsma.nl。URL: https://markkamsma.nl/phd-thesis。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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