2021/12/02 by Samson Leung, Leung, Samson
Mathematics · #03C48 (Primary) 03C45 #03C55 (Secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2112.01572
openalex publication_date 2021/12/02 · openalex created_date 2021/12/31 · openalex updated_date 2026/07/28
Assuming the existence of a monster model, tameness and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let μ>LS(\bf K) be a regular stability cardinal and let χ be the local character of μ-nonsplitting. The following holds: 1. When μ-nonforking is restricted to (μ,≥χ)-limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension and continuity. It also has local character χ. This generalizes Vasey's result which assumed μ-superstability to obtain same properties but with local character ℵ0. 2. There is λ∈[μ,h(μ)) such that if \bf K is stable in every cardinal between μ and λ, then \bf K has μ-symmetry while μ-nonforking in (1) has symmetry. In this case (a) \bf K has the uniqueness of (μ,≥χ)-limit models: if M1,M2 are both (μ,≥χ)-limit over some M0∈ Kμ, then M1≅M0M2; (b) any increasing chain of μ+-saturated models of length ≥χ has a μ+-saturated union. These generalize VanDieren-Vasey's result and remove the symmetry assumption in Boney-VanDieren and Vasey's result. Under (