Long limit models are isomorphic assuming a splitting-like relation

TL;DR

This paper proves the uniqueness of high cofinality limit models in stable AECs under weak independence assumptions, extending previous results to broader classes of independence relations and limit models.

Contribution

It introduces a generalized framework for the uniqueness of limit models in stable AECs using weaker independence relations, broadening the scope of prior results.

Findings

Uniqueness of high cofinality limit models under weak independence assumptions.

Extension of previous theorems to broader classes of independence relations.

Generalization of all known positive isomorphism results for limit models.

Abstract

We prove the uniqueness of high cofinality limit models in stable abstract elementary classes (AECs) with amalgamation, assuming the existence of a rather weak independence relation. $\textbf{Theorem.}$ Suppose $\mathbf{K}$ is a $\lambda$-stable AEC, where $\operatorname{LS}(\mathbf{K}) \leq \lambda$, $\kappa < \lambda^+$ is regular, and $\mathbf{K}_\lambda$ satisfies the amalgamation property. Let $\mathbf{K}'$ is the class of all $(\lambda, \delta)$-limit models where $\operatorname{cf}(\delta) \geq \kappa$ (or any AC where $\mathbf{K}' \subseteq \mathbf{K}_\lambda$ contains all such $(\lambda, \delta)$-limit models when $\operatorname{cf}(\delta) \geq \kappa$). Suppose also that there is an independence relation on $\mathbf{K}'$ satisfying weak uniqueness, weak existence, universal continuity* in $\mathbf{K}_\lambda$, $(\geq \kappa)$-local character, and $(\lambda, \theta)$-weak non-forking amalgamation in some regular $\theta \in [\kappa, \lambda^+)$. Let $\delta_1, \delta_2 < \lambda^+$ be limit with $\operatorname{cf}(\delta_l) \geq \kappa$ for $l = 1, 2$. Then for all $M, N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit over $M$ for $l = 1, 2$, then $N_1 \underset{M}{\cong} N_2$. Moreover, if $K_\lambda$ also satisfies the joint embedding property, then for all $N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit for $l = 1, 2$, then $N_1 {\cong} N_2$. This generalises both Theorem 3.1 of arXiv:2503.11605 and Theorem 1.2 of arXiv:1508.04717 - the former to apply to independence relations that satisfy much weaker forms of uniqueness, extension, and non-forking amalagamation, and the latter to independence relations other than $\lambda$-non-splitting. As such, this generalises all other positive isomorphism results of limit models known to the author.