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Long limit models are isomorphic assuming a splitting-like relation

Jeremy Beard

TL;DR

The paper addresses when long limit models in a $oldsymbol{K}$, a $oldsymbol{lambda}$-stable AEC with amalgamation, are isomorphic. It develops a towers framework—comprising universal and reduced towers and brilliant chains—to prove that two limit models of lengths $oldsymbol{delta_1}$ and $oldsymbol{delta_2}$ over a base $oldsymbol{M}$ are isomorphic given a weak independence relation with weak uniqueness, weak existence, universal continuity*, and $(oldsymbol{lambda},oldsymbol{theta})$-weak non-forking amalgamation. The main theorem generalizes previous results (bovan and bema) by allowing weaker forms of standard independence properties and by avoiding the monster-model assumption. The constructions ensure the final tower is universal and continuous at the relevant cofinalities, yielding a model that simultaneously realizes both limit conditions. These results have broad implications, recovering known isomorphism theorems and extending them to a wider class of stable AECs and independence notions, thereby enriching the understanding of the isomorphism spectrum of limit models in non-elementary contexts.

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 $λ$-stable AEC, where $\operatorname{LS}(\mathbf{K}) \leq λ$, $κ< λ^+$ is regular, and $\mathbf{K}_λ$ satisfies the amalgamation property. Let $\mathbf{K}'$ is the class of all $(λ, δ)$-limit models where $\operatorname{cf}(δ) \geq κ$ (or any AC where $\mathbf{K}' \subseteq \mathbf{K}_λ$ contains all such $(λ, δ)$-limit models when $\operatorname{cf}(δ) \geq κ$). Suppose also that there is an independence relation on $\mathbf{K}'$ satisfying weak uniqueness, weak existence, universal continuity* in $\mathbf{K}_λ$, $(\geq κ)$-local character, and $(λ, θ)$-weak non-forking amalgamation in some regular $θ\in [κ, λ^+)$. Let $δ_1, δ_2 < λ^+$ be limit with $\operatorname{cf}(δ_l) \geq κ$ for $l = 1, 2$. Then for all $M, N_1, N_2 \in \mathbf{K}_λ$, if $N_l$ is $(λ, δ_l)$-limit over $M$ for $l = 1, 2$, then $N_1 \underset{M}{\cong} N_2$. Moreover, if $K_λ$ also satisfies the joint embedding property, then for all $N_1, N_2 \in \mathbf{K}_λ$, if $N_l$ is $(λ, δ_l)$-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 $λ$-non-splitting. As such, this generalises all other positive isomorphism results of limit models known to the author.

Long limit models are isomorphic assuming a splitting-like relation

TL;DR

The paper addresses when long limit models in a , a -stable AEC with amalgamation, are isomorphic. It develops a towers framework—comprising universal and reduced towers and brilliant chains—to prove that two limit models of lengths and over a base are isomorphic given a weak independence relation with weak uniqueness, weak existence, universal continuity*, and -weak non-forking amalgamation. The main theorem generalizes previous results (bovan and bema) by allowing weaker forms of standard independence properties and by avoiding the monster-model assumption. The constructions ensure the final tower is universal and continuous at the relevant cofinalities, yielding a model that simultaneously realizes both limit conditions. These results have broad implications, recovering known isomorphism theorems and extending them to a wider class of stable AECs and independence notions, thereby enriching the understanding of the isomorphism spectrum of limit models in non-elementary contexts.

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. Suppose is a -stable AEC, where , is regular, and satisfies the amalgamation property. Let is the class of all -limit models where (or any AC where contains all such -limit models when ). Suppose also that there is an independence relation on satisfying weak uniqueness, weak existence, universal continuity* in , -local character, and -weak non-forking amalgamation in some regular . Let be limit with for . Then for all , if is -limit over for , then . Moreover, if also satisfies the joint embedding property, then for all , if is -limit for , then . 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 -non-splitting. As such, this generalises all other positive isomorphism results of limit models known to the author.

Paper Structure

This paper contains 17 sections, 32 theorems, 5 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Limit models
  5. Independence relations
  6. On Symmetry
  7. Main result
  8. An outline of the proof
  9. Comments on non-algebraic types, JEP, and NMM
  10. Non-algebraic independence relations, and why we won't use them
  11. JEP and NMM, and why we could assume them if we wanted to
  12. Towers and brilliant chains
  13. Extending towers and brilliant chains
  14. Full towers
  15. Reduced chains of towers
  16. ...and 2 more sections

Key Result

Lemma 2.12

Suppose $\mathbf{K}$ is an AEC and $\lambda \geq \operatorname{LS}(\mathbf{K})$, that $\mathbf{K}$ is $\lambda$-stable and $\mathbf{K}_\lambda$ has AP, and that $\mathbf{K}'$ is an AC where $\mathbf{K}_{(\lambda, \geq \kappa)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_\lambda$. Then

Figures (8)

  • Figure 3: The application of symmetry in Lemma \ref{['symmetry_implies_nfap']}
  • Figure 4: The construction in Lemma \ref{['symmetry_implies_nfap']}
  • Figure 5: The chain we will construct, looking only at models for rows in the 'copy of $\delta_2$'
  • Figure 6: A $\vartriangleleft$-chain of length $\alpha+1$. Note the singletons appear two levels up in the tower, and we may only insert new levels after an infinite chain of models because of condition (5) of \ref{['tower_ordering_def']}. $\mathcal{T}^0, \mathcal{T}^1$, and $\mathcal{T}^2$ are ordered by $\omega + 4$, whereas $\mathcal{T}^3, \mathcal{T}^4$, and so on are ordered by $\{0, 1, 2, 3, \dots, \frac{\omega}{2}, \frac{\omega}{2} + 1, \frac{\omega}{2} + 2, \dots, \omega, \omega+1, \omega+2, \omega+3\}$. That is, we inserted $\omega$-many new levels just under level $\omega$.
  • Figure 7: The successor case of Proposition \ref{['univ-tower-exts-exist']}
  • ...and 3 more figures

Theorems & Definitions (108)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9: sh394
  • Definition 2.11
  • Lemma 2.12
  • proof
  • ...and 98 more