Limit Models in Strictly Stable Abstract Elementary Classes

2015/08/19 by Will Boney, Boney, Will, Monica VanDieren +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1508.04717

Abstract

In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note that no tameness is assumed): Suppose that K is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality μ. 2. stability in μ. 3. κ^*_μ(K)<μ⁺. 4. continuity for non-μ-splitting (i.e. if p\inga-S(M) and M is a limit model witnessed by ⟨ M_i| i<α⟩ for some limit ordinal α<μ⁺ and there exists N \prec M₀ so that p\restriction M_i does not μ-split over N for all i<α, then p does not μ-split over N). For θ and δ limit ordinals <μ⁺ both with cofinality ≥κ^*_μ(K), if K satisfies symmetry for non-μ-splitting (or just (μ,δ)-symmetry), then, for any M₁ and M₂ that are (μ,θ) and (μ,δ)-limit models over M₀, respectively, we have that M₁ and M₂ are isomorphic over M₀.

Cited by

• Long limit models are isomorphic assuming a splitting-like relation
• The spectrum of limit models in a first order setting

Related

• On the spectrum of limit models
• Uniqueness of Limit Models in Classes with Amalgamation
• Uniqueness of Limit Models in Classes with Amalgamation
• Long limit models are isomorphic assuming a splitting-like relation
• Superstability in Tame Abstract Elementary Classes