Superstability and Symmetry

Monica M. VanDieren

Published 2015-07-07Version 1

In this paper we extend the work of \cite{ShVi}, \cite{Va1}, \cite{Va2}, and \cite{GVV} by weakening the structural assumptions on $\mathcal{K}_{\mu^+}$ to something more closely resembling superstability from first order logic in order to derive the uniqueness of limit models of cardinality $\mu$. Moreover, we identify a necessary and sufficient condition for symmetry of non-splitting that involves reduced towers. This new condition is particularly interesting since it does not have a pre-established first-order analog. Additionally, this condition provides a mechanism for deriving symmetry in abstract elementary classes without having to assume set-theoretic assumptions or tameness: Corollary: Suppose that $\mathcal{K}$ satisfies the amalgamation and joint embedding properties and $\mu$ is a cardinal $\geq\beth_{(2^{Hanf(\mathcal{K})})^+}$. If $\mathcal{K}$ is categorical in $\lambda=\mu^+$, then $\mathcal{K}$ has symmetry for non-$\mu$-splitting.