Limit Models in Strictly Stable Abstract Elementary Classes

TL;DR

This paper investigates the conditions under which limit models in certain stable abstract elementary classes are unique up to isomorphism, focusing on the locality of non-splitting without assuming tameness.

Contribution

It establishes the level of uniqueness of limit models in stable, non-superstable AECs under locality and symmetry conditions without requiring tameness.

Findings

Proves isomorphism of limit models under specified conditions.

Shows continuity for non-splitting in stable AECs.

Demonstrates uniqueness results without tameness assumptions.

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 $\mathcal{K}$ is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $\mu$. 2. stability in $\mu$. 3. $\kappa^*_\mu(\mathcal{K})<\mu^+$. 4. continuity for non-$\mu$-splitting (i.e. if $p\in\text{ga-S}(M)$ and $M$ is a limit model witnessed by $\langle M_i\mid i<\alpha\rangle$ for some limit ordinal $\alpha<\mu^+$ and there exists $N \prec M_0$ so that $p\restriction M_i$ does not $\mu$-split over $N$ for all $i<\alpha$, then $p$ does not $\mu$-split over $N$). For $\theta$ and $\delta$ limit ordinals $<\mu^+$ both with cofinality $\geq\kappa^*_\mu(\mathcal{K})$, if $\mathcal{K}$ satisfies symmetry for non-$\mu$-splitting (or just $(\mu,\delta)$-symmetry), then, for any $M_1$ and $M_2$ that are $(\mu,\theta)$ and $(\mu,\delta)$-limit models over $M_0$, respectively, we have that $M_1$ and $M_2$ are isomorphic over $M_0$.