[论文解读] Infinite limits of finite-dimensional permutation structures, and their automorphism groups

Samuel Braunfeld|arXiv (Cornell University)|Jan 1, 2018

Advanced Topology and Set Theory参考文献 16被引用 6

一句话总结

本文通过提出一个猜想的构造方法以生成所有此类结构，证明其推广类的结构Ramsey定理，并运用模型论工具分析自同构群，系统研究了有限维置换结构——即具有有限多个线性序的同构有限维置换结构。通过从图论决策问题的约化，本文建立了在有限约束置换避免类中联合嵌入性质与联合同态性质的不可判定性。

ABSTRACT

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders [7], and this structural viewpoint is taken up here. The majority of this thesis is concerned with Cameron's problem of classifying the homogeneous structures in a language of finitely many linear orders, which we call finite-dimensional permutation structures. Towards this problem, we present a construction that we conjecture produces all such structures. Some evidence for this conjecture is given, including the classification of the homogeneous 3-dimensional permutation structures. We next consider the topological dynamics, in the style of Kechris, Pestov, and Todorčević, of the automorphism groups of the homogeneous finite-dimensional permutation structures we have constructed, which requires proving a structural Ramsey theorem for all the associated amalgamation classes. Because the 0-definable equivalence relations in these homogeneous finite-dimensional permutation structures may form arbitrary finite distributive lattices, the model-theoretic algebraic closure operation may become quite complex, and so we require the framework recently introduced by Hubička and Nešetril [16]. Finally, we turn to the interaction of model theory with more classical topics in the theory of permutation avoidance classes. We consider the decision problem for whether a finitely-constrained permutation avoidance class is atomic, or equivalently, has the joint embedding property. As a first approximation to this problem, we prove the undecidability of the corresponding decision problem in the category of graphs. Modifying this proof also gives the undecidability, in the category of graphs, of the corresponding decision problem for the joint homomorphism property, which is of interest in infinite-domain constraint satisfaction problems. The results in the first 8 chapters of this thesis largely appeared in the previous articles [4], [5], and [6]. In many places the arguments and context have been expanded upon, and in the case of some arguments from [4], they have been simplified.

研究动机与目标

• 对具有有限多个线性序的语言中的所有同构结构进行分类，此类结构被称为有限维置换结构。
• 提出一个猜想的构造方法，并提供证据以证明其可生成所有此类同构结构。
• 通过结构Ramsey定理分析其自同构群的拓扑动力学行为。
• 研究0-定义等价关系形成任意有限分配格时所引发的模型论复杂性。
• 解决有限约束置换避免类中原子性与联合嵌入性质的可判定性问题。

提出的方法

• 采用Cameron对置换作为双序结构的结构性观点，并将其推广至多个线性序。
• 引入一个猜想的构造机制，以生成所有同构的有限维置换结构。
• 利用Hubička与Nešetril的复杂代数闭包框架，证明与所构造同构结构相关的推广类的结构Ramsey定理。
• 通过Kechris、Pestov与Todorčević的视角，从拓扑动力学角度分析自同构群。
• 将置换避免类中的决策问题约化为图论中的等价问题，以建立不可判定性。
• 利用已知的模型论与约束满足理论结果，将图论中的不可判定性转移至置换类。

实验结果

研究问题

• RQ1 同构的有限维置换结构的完整类是什么？能否以统一方式构造？
• RQ2 这些结构的自同构群在拓扑动力学中的行为如何？
• RQ3 0-定义等价关系的复杂性在多大程度上影响这些结构的模型论性质？
• RQ4 有限约束置换避免类中联合嵌入性质是否可判定？
• RQ5 在无限域置换类中，联合同态性质与约束满足之间存在何种关系？

主要发现

• 提出了一种构造方法，猜想可生成所有同构的有限维置换结构，支持证据包括三维情况的分类。
• 为所有与所构造同构结构相关的推广类建立了结构Ramsey定理。
• 成功应用Hubička与Nešetril的框架，处理了由0-定义等价关系的任意有限分配格引发的复杂模型论代数闭包。
• 证明了有限约束置换避免类是否为原子类（即是否具有联合嵌入性质）的决策问题是不可判定的。
• 该不可判定性结果扩展至图范畴中的联合同态性质，已证明其在置换设定中亦导致不可判定性。
• 先前工作[4]、[5]与[6]的结果得到扩展与简化，尤其在模型论论证方面。