Limit models in strictly stable abstract elementary classes

• Published in: Mathematical logic quarterly Vol. 70; no. 4; pp. 438 - 453
• Main Authors: Boney, Will, VanDieren, Monica M.
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.11.2024
• Subjects: Splitting; Symmetry
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202200075

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, elementary classes. In particular we prove the following. Suppose that K$\mathcal {K}$ is an elementary class satisfying (1)the joint embedding and amalgamation properties with no maximal model of cardinality μ$\mu$, (2)stability in μ$\mu$, (3)κμ∗(K)<μ+$\kappa ^*_\mu (\mathcal {K})<\mu ^+$, (4)continuity for non-μ-splitting${\rm non}\text{-}\mu\text{-}{\rm splitting}$ (i.e., if p∈ga-S(M)$p\in \operatorname{ga-S}(M)$ and M$M$ is a limit model witnessed by ⟨Mi|i<α⟩$\langle M_i| i<\alpha \rangle$ for some limit ordinal α<μ+$\alpha <\mu ^+$ and there exists N≺M0$N \prec M_0$ so that p↾Mi$p\mathord {\upharpoonright }M_i$ does not μ$\mu$‐split over N$N$ for all i<α$i<\alpha$, then p$p$ does not μ$\mu$‐split over N$N$). Then for ϑ$\vartheta$ and δ$\delta$ limit ordinals <μ+$<\mu ^+$ both with cofinality ≥κμ∗(K)$\ge \kappa ^*_\mu (\mathcal {K})$, if K$\mathcal {K}$ satisfies symmetry for non-μ-splitting${\rm non}\text{-}\mu\text{-}{\rm splitting}$ (or just (μ,δ)$(\mu,\delta)$‐symmetry), then, for any M1$M_1$ and M2$M_2$ that are (μ,ϑ)$(\mu,\vartheta)$ and (μ,δ)$(\mu,\delta)$‐limit models over M0$M_0$, respectively, we have that M1$M_1$ and M2$M_2$ are isomorphic over M0$M_0$. Note that no tameness is assumed.