Transferring symmetry downward and applications

Monica M. VanDieren, Sebastien Vasey

Published 2015-08-13Version 1

For $K$ an abstract elementary class satisfying the amalgamation property, we prove a downward transfer of the symmetry property for splitting (previously isolated by the first author). This allows us to deduce uniqueness of limit models from categoricity in a cardinal of high-enough cofinality, improving on a 16-year-old result of Shelah: $\mathbf{Theorem}$ Suppose $\lambda$ and $\mu$ are cardinals so that $\text{cf}(\lambda)>\mu\geq\text{LS}(K)$ and assume that $K$ has no maximal models and is categorical in $\lambda$. If $M_0,M_1,M_2 \in K_\mu$ are such that both $M_1$ and $M_2$ are limit models over $M_0$, we have that $M_1\cong_{M_0}M_2$. Another application of the symmetry transfer utilizes tameness (a locality property for types) and improves on the work of Will Boney and the second author: $\mathbf{Theorem}$ Let $\mu \ge \text{LS} (K)$. If $K$ is $\mu$-superstable and $\mu$-tame, then: * If $M_0, M_1, M_2 \in K_\mu$ are such that both $M_1$ and $M_2$ are limit models over $M_0$, then $M_1 \cong_{M_0} M_2$. * For any $\lambda > \mu$, the union of an increasing chain of $\lambda$-saturated models is $\lambda$-saturated. * There exists a unique type-full good $\mu^+$-frame with underlying class the saturated models in $K_{\mu^+}$.