[Paper Review] Infinite limits of finite-dimensional permutation structures, and their automorphism groups

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

Advanced Topology and Set Theory16 references6 citations

TL;DR

This paper investigates homogeneous finite-dimensional permutation structures—structures with finitely many linear orders—by proposing a construction conjectured to generate all such structures, proving a structural Ramsey theorem for their amalgamation classes, and applying model-theoretic tools to analyze automorphism groups. It establishes the undecidability of the joint embedding and joint homomorphism properties in finitely-constrained permutation avoidance classes via reductions from graph-theoretic decision problems.

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.

Motivation & Objective

• To classify all homogeneous structures in a language of finitely many linear orders, known as finite-dimensional permutation structures.
• To propose and provide evidence for a conjectured construction that generates all such homogeneous structures.
• To analyze the topological dynamics of their automorphism groups using a structural Ramsey theorem.
• To investigate the model-theoretic complexity arising from 0-definable equivalence relations forming arbitrary finite distributive lattices.
• To address the decidability of atomicity and the joint embedding property in finitely-constrained permutation avoidance classes.

Proposed method

• Adopting Cameron’s structural viewpoint of permutations as two-order structures, extending it to multiple linear orders.
• Introducing a conjectured construction mechanism for generating all homogeneous finite-dimensional permutation structures.
• Proving a structural Ramsey theorem for the associated amalgamation classes using the framework of Hubička and Nešetril for complex algebraic closure.
• Analyzing automorphism groups through the lens of topological dynamics, following Kechris, Pestov, and Todorčević.
• Reducing decision problems in permutation avoidance classes to equivalent problems in graph theory to establish undecidability.
• Leveraging known results from model theory and constraint satisfaction to transfer undecidability from graphs to permutation classes.

Experimental results

Research questions

• RQ1 What is the complete class of homogeneous finite-dimensional permutation structures, and can they be uniformly constructed?
• RQ2 How do the automorphism groups of these structures behave in terms of topological dynamics?
• RQ3 To what extent does the complexity of 0-definable equivalence relations affect the model-theoretic properties of these structures?
• RQ4 Is the joint embedding property decidable for finitely-constrained permutation avoidance classes?
• RQ5 What is the relationship between the joint homomorphism property and constraint satisfaction in infinite-domain permutation classes?

Key findings

• A construction is proposed that is conjectured to generate all homogeneous finite-dimensional permutation structures, with supporting evidence including the classification of the 3-dimensional case.
• A structural Ramsey theorem is established for all amalgamation classes associated with the constructed homogeneous structures.
• The framework of Hubička and Nešetril is successfully applied to handle the complex model-theoretic algebraic closure arising from arbitrary finite distributive lattices of 0-definable equivalence relations.
• The decision problem for whether a finitely-constrained permutation avoidance class is atomic (i.e., has the joint embedding property) is proven undecidable.
• The undecidability result extends to the joint homomorphism property in the category of graphs, which is shown to imply undecidability in the permutation setting.
• The results from earlier works [4], [5], and [6] are expanded and simplified, particularly in the model-theoretic arguments.