2025/11/24 by Beard, Jeremy
Computer Science · Mathematics · #03C50 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems #Primary: 03C48. Secondary: 03C45
paper · doi:10.48550/arxiv.2511.18665
openalex publication_date 2025/11/24 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28
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. Theorem. Suppose K is a λ-stable AEC, where LS(K) ≤ λ, κ< λ+ is regular, and Kλ satisfies the amalgamation property. Let K' is the class of all (λ, δ)-limit models where cf(δ) ≥ κ (or any AC where K' ⊆ Kλ contains all such (λ, δ)-limit models when cf(δ) ≥ κ). Suppose also that there is an independence relation on K' satisfying weak uniqueness, weak existence, universal continuity* in Kλ, (≥ κ)-local character, and (λ, θ)-weak non-forking amalgamation in some regular θ∈ [κ, λ+). Let δ1, δ2 < λ+ be limit with cf(δl) ≥ κ for l = 1, 2. Then for all M, N1, N2 ∈ Kλ, if Nl is (λ, δl)-limit over M for l = 1, 2, then N1 \undersetM≅ N2. Moreover, if Kλ also satisfies the joint embedding property, then for all N1, N2 ∈ Kλ, if Nl is (λ, δl)-limit for l = 1, 2, then N1 ≅ N2. 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.