Long limit models are isomorphic assuming a splitting-like relation
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2025
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| _version_ | 1866908671241355264 |
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| author | Beard, Jeremy |
| author_facet | Beard, Jeremy |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_18665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long limit models are isomorphic assuming a splitting-like relation Beard, Jeremy Logic Primary: 03C48. Secondary: 03C45, 03C50 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. |
| title | Long limit models are isomorphic assuming a splitting-like relation |
| topic | Logic Primary: 03C48. Secondary: 03C45, 03C50 |
| url | https://arxiv.org/abs/2511.18665 |