Transferring Symmetry

Abstract

In this paper, we apply results of \cite{Va3} and use towers to transfer symmetry from ${\mu }^{+}$ down to $\mu$ in superstable abstract elementary classes without using extra set-theoretic assumptions or tameness. Theorem. Suppose $\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\mathcal{K}$ is both $\mu$- and ${\mu }^{+}$-superstable. If $\mathcal{K}$ has symmetry for non-${\mu }^{+}$-splitting, then $\mathcal{K}$ has symmetry for non-$\mu$-splitting. This is a new application of towers which were introduced by Shelah and Villaveces \cite{ShVi} and later used by VanDieren \cite{Va1}, \cite{Va2} and Grossberg, VanDieren, and Villaveces \cite{GVV} to prove the uniqueness of limit models.

This paper has been combined with another paper. The content appears in arXiv:1512.01786

Cited by in corpus (2)

• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes