Core Concepts

This research paper investigates the structural properties of two specific types of infinite permutation groups, those preserving limits of betweenness relations and those preserving limits of D-relations, revealing their complexity and significance in model theory.

Abstract

**Bibliographic Information:**Almazaydeh, A. I., Braunfeld, S., & Macpherson, D. (2024). Omega-categorical limits of betweenness relations and $D$-sets. arXiv preprint arXiv:2410.05832v1.**Research Objective:**This paper aims to address open questions regarding the structural and model-theoretic properties of two constructions of oligomorphic Jordan permutation groups preserving a ‘limit of betweenness relations’ (MB) and a ‘limit of D-relations’ (MD) introduced in previous work.**Methodology:**The authors utilize concepts and techniques from model theory and permutation group theory, including Fraïssé amalgamation, indiscernible sequences, and analysis of growth rates of orbits. They examine the homogeneity, model-theoretic properties (NIP, monadic NIP, age), and maximality of the automorphism groups of MB and MD.**Key Findings:**- Both MB and MD are not homogeneous in their original languages but are homogenizable, meaning they can be made homogeneous by adapting the language while preserving the automorphism group.
- Both structures are NIP (not having the independence property), implying a certain level of tameness in their model-theoretic behavior.
- However, both MB and MD are not monadically NIP, their ages are not well-quasi-ordered under embeddability, and the growth rate of orbits on k-sets grows faster than exponentially.
- The automorphism groups of both MB and MD are maximal-closed in the symmetric group, indicating that these structures do not have any proper non-trivial reducts.

**Main Conclusions:**The study reveals that while MB and MD exhibit some degree of model-theoretic tameness (being NIP), they are structurally complex, as evidenced by their non-homogeneity, lack of monadic NIP, and super-exponential orbit growth. The maximality of their automorphism groups further underscores their significance in the study of Jordan groups.**Significance:**This research contributes to a deeper understanding of the model theory and permutation group theory of infinite structures, particularly within the class of Jordan groups. The intricate constructions of MB and MD and their properties provide valuable insights into the behavior of limits of betweenness relations and D-sets.**Limitations and Future Research:**The paper primarily focuses on specific model-theoretic and group-theoretic aspects of MB and MD. Further research could explore additional model-theoretic properties, such as dp-minimality and distality, and investigate potential applications of these structures as counter-examples in other areas of model theory and permutation group theory.

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by Asma Ibrahim... at **arxiv.org** 10-10-2024

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While the structures MB and MD themselves exhibit super-exponential orbit growth, their connection to other omega-categorical structures with non-super-exponential growth lies in their building blocks: the basic treelike structures. Here's a breakdown:
Shared Foundation: Both MB and MD are constructed from families of betweenness relations and D-relations, which are examples of basic treelike structures. These structures, when considered in isolation, can exhibit non-super-exponential orbit growth. For instance, specific examples of homogeneous expansions of C-relations and D-relations, such as those equipped with compatible linear or circular orders, have been shown to have this property ([15]).
Crucial Difference: Limits and Intertwining: The key distinction arises from the "limit" nature of MB and MD. They are not simply betweenness relations or D-relations but intricate structures where these basic treelike components are intertwined across different levels indexed by the semilinear order (J, ≤). This intertwining, facilitated by the maps fi and gij, is crucial for encoding the complexity that leads to super-exponential orbit growth.
Analogy: Building Complex Structures: Think of it like this: a single line segment has a simple structure. However, if you arrange infinitely many line segments in a carefully defined hierarchical way, with connections between segments at different levels, you can create incredibly complex shapes like fractals. Similarly, the "limit" construction in MB and MD takes relatively simple, potentially non-super-exponential, basic treelike structures and assembles them into a structure with much higher complexity, resulting in super-exponential orbit growth.

It's highly unlikely that the constructions of MB and MD could be modified to be monadically NIP while preserving their fundamental nature as limits of treelike structures. Here's why:
Source of IP: Treelike Structure and Limits: The independence property in MB and MD is deeply rooted in the treelike nature of their constituent betweenness relations and D-relations, and amplified by the limit construction. The branching nature of these structures inherently allows for the coding of arbitrarily long linear orders, a hallmark of IP. The limit construction further strengthens this by propagating the IP across infinitely many levels.
Modifications Would Likely Break the Structure: Attempting to eliminate IP while retaining the essence of MB and MD would likely require fundamental changes that undermine their intended purpose. For example:
Restricting Branching: Limiting the branching in the underlying treelike structures could potentially control IP, but it would also drastically alter the fundamental properties of betweenness relations and D-relations.
Finite Levels: Truncating the structures to have finitely many levels in J would remove the infinite propagation of IP. However, this would no longer capture the notion of a "limit" of treelike structures.
Orbit Growth Implications: If, hypothetically, a monadically NIP variant of MB or MD were to be constructed, it would, by Conjecture 1 from [14], necessarily have at most exponential orbit growth. This underscores the tight connection between model-theoretic properties (like NIP and monadic NIP) and combinatorial properties (like orbit growth) in omega-categorical structures.

The maximality of the automorphism groups of MB and MD, as established in Theorem 1.4, has significant implications for the classification of closed oligomorphic primitive Jordan groups. Here's how:
Ruling Out Reducts: The maximality implies that MB and MD have no proper non-trivial reducts. In the context of classifying Jordan groups, this means that we cannot obtain new examples of such groups by simply taking reducts of MB or MD. This simplifies the classification task by eliminating a potential source of new examples.
Understanding Constraints: The fact that the rich structure of MB and MD cannot be further reduced suggests that the combination of 2-primitivity, 3-homogeneity, and the specific Jordan set properties described in Propositions 2.3 and 2.4 imposes very strong constraints on the possible permutation groups that can arise.
Towards a More Precise Classification: This knowledge can potentially contribute to a more precise classification of closed oligomorphic primitive Jordan groups. By understanding the limits of reducibility in these key examples, we gain a better grasp of the possible structures and their corresponding automorphism groups within this class.
Open Questions and Future Directions: While the maximality result is a significant step, it also raises further questions:
Characterizing Maximal-Closed Jordan Groups: Can we characterize all closed oligomorphic primitive Jordan groups that are maximal-closed? What group-theoretic or combinatorial properties guarantee this maximality?
Beyond MB and MD: Are there other "canonical" structures for limits of betweenness relations or D-relations that might admit proper reducts? Exploring these questions could lead to a more complete understanding of the landscape of closed oligomorphic primitive Jordan groups.

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