Elementary submodels in infinite combinatorics

A quotient-like construction involving elementary submodels

This article is an investigation of a method of deriving a topology from a space and an elementary submodel containing it. We first define and give the basic properties of this construction, known as $X/M$. In the next section, we construct some examples and analyse the topological relationship between $X$ and $X/M$. In the final section, we apply $X/M$ to get novel results about Lindelöf spaces, giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

关于初等子模型的下向Löwenheim-Skolem定理

On the downward Löwenheim-Skolem Theorem for elementary submodels

本文通过基于形式系统中替换的新模型定义，提出了一种纯代数方法，并给出了初等子模型的下向Löwenheim-Skolem定理的新证明。

Elementary submodels, coding strategies, and an infinite real number game

Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique from infinite game theory, may be used to answer Baker's question on whether this game provides a precise characterization for countable subsets of real numbers, and we connect this game to the well-studied Banach-Mazur game from topology.

逼近理论与初等子模型

Approximation Theory and Elementary Submodels

该研究运用集合论初等子模型方法，为逼近理论中的知名定理给出新的简短证明，部分结果更强，丰富了逼近理论的证明手段。

权重、净权重与初等子模型

Weight, net weight, and elementary submodels

本文通过初等子模型方法，在ZFC下构造了$C(ω_1)$中净权重不可数的Hausdorff空间，并证明添加Cohen实数后正则空间净权重可数，同时解决了文献[6]中的两个主要问题。

Stably embedded submodels of Henselian valued fields

We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued fields is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field $\mathbb{R}((\mathbb{Z}))$ are definable. Similarly, all types over the quotient field of the Witt…

弱反映图性质

Weakly reflecting graph properties

本文研究 L. Soukup 用于不可数图的初等子模型框架，探讨将“良反映”性质假设弱化为自然出现的较弱形式，以理解相关无限图论问题。

Strong splitting in stable homogeneous models

We study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in [T. Hyttinen, On nonstructure of elementary submodels of a stable homogeneous structure, Fundamenta Mathematicae, 156(1998): 167-182]. We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure.

$\mathsf{PA}$ 模型的格问题

The Lattice Problem for Models of $\mathsf{PA}$

本文综述了 Peano 算术模型格问题 60 年的研究进展，详细分析了特殊情况以解释 James Schmerl 构造具有指定结构间格初等扩张的技术，并讨论了可数递归饱和模型的初等子模型格的相关结果。

非初等范畴性与射影局部 o-极小类

Non-elementary categoricity and projective locally o-minimal classes

本文将光滑复代数簇族的覆盖与实数 o-极小扩张中局部可定义的结构类 $\mathcal{U}$ 相关联，证明了其子模型上的 $\aleph_0$-齐次性、稳定性及 $\aleph_1$-范畴性，并在一维情形下修改得到所有不可数基数下范畴的抽象初等类。

Introduction to: classification theory for abstract elementary class

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to find dividing lines, prove their consequences, prove "structure theorems, positive theorems" on those in the "low side" (in particular stable and superstable theories), and prove "non-structure, complexity theorems" on the "high side". It has started with categoricity and number of non-isomorphic models. It is probably recognized as the central part of model …

Abstract decomposition theorem and applications

Let K be an Abstract Elementary Class. Under the asusmptions that K has a nicely behaved forking-like notion, regular types and existence of some prime models we establish a decomposition theorem for such classes. The decomposition implies a main gap result for the class K. The setting is general enough to cover \aleph_0-stable first-order theories (proved by Shelah in 1982), Excellent Classes of atomic models of a first order tehory (proved Grossberg and Hart 1987) and the class of submodels of a large sequentially homogenuus \aleph_0-stable model (which is new).

作为项的多项式与布尔独立性定理

Polynomials as terms and the Boolean Independence Theorem

本文将交换环上的形式多元多项式视为环项，证明 s-等价与 f-等价等价，并类似地建立布尔项的独立性定理，为概率方法论证提供严格基础。