2015/09/04 by Monica VanDieren, VanDieren, Monica M., Sebastien Vasey +1
Computer Science · Mathematics · #03C45 #03C48 (Primary) #03C52 #03C55 (Secondary) #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1509.01488
openalex publication_date 2015/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For K an abstract elementary class with amalgamation and no maximal models,\nwe show that categoricity in a high-enough cardinal implies structural\nproperties such as the uniqueness of limit models and the existence of good\nframes. This improves several classical results of Shelah.\n \Theorem\n Let \μ \≥ \LS (K). If K is categorical in a \λ \≥\n beth\(2\μ\)+, then:\n 1) Whenever M0, M1, M2 \∈ K_\μ are such that M1 and M2 are limit\nover M0, we have M1 \≅M0 M2.\n 2) If \μ > \LS (K), the model of size \λ is \μ-saturated.\n 3) If \μ \≥ beth(2\LS (K))+ and \λ \≥\n beth\(2\μ+\)+, then there exists a type-full good\n\μ-frame with underlying class the saturated models in K_\μ.\n Our main tool is the symmetry property of splitting (previously isolated by\nthe first author). The key lemma deduces symmetry from failure of the order\nproperty.\n