Neostability in countable homogeneous metric spaces

TL;DR

This paper studies the model-theoretic properties of a class of universal metric spaces derived from ordered monoids, revealing how algebraic features influence properties like stability and simplicity.

Contribution

It characterizes model-theoretic properties of $ ext{Th}(U_R)$ using algebraic properties of the monoid $R$, connecting algebra and model theory.

Findings

Characterizes stability, simplicity, and SOP$_n$ hierarchy via algebraic properties of $R$.

Provides criteria for superstability, supersimplicity, and elimination of imaginaries.

Establishes necessary conditions for elimination of hyperimaginaries.

Abstract

Given a countable, totally ordered commutative monoid $\mathcal{R}=(R,\oplus,\leq,0)$, with least element $0$, there is a countable, universal and ultrahomogeneous metric space $\mathcal{U}_\mathcal{R}$ with distances in $\mathcal{R}$. We refer to this space as the $\mathcal{R}$-Urysohn space, and consider the theory of $\mathcal{U}_\mathcal{R}$ in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free $n^{\text{th}}$ roots of the complete graph (e.g. the random graph when $n=2$), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of $\text{Th}(\mathcal{U}_\mathcal{R})$ by algebraic properties of $\mathcal{R}$, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOP$_n$-hierarchy. Using the submonoid of idempotents in $\mathcal{R}$, we also characterize superstability, supersimplicity, and weak elimination of imaginaries. Finally, we give necessary conditions for elimination of hyperimaginaries, which further develops previous work of Casanovas and Wagner.