Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes

Our main result (Theorem 1) suggests a possible dividing line ($\mu$-superstable $+$ $\mu$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theoerem 1: Let $\mathcal{K}$ be an abstract elementary class with no maximal models of cardinality ${\mu }^{+}$ which satisfies the joint embedding and amalgamation properties. Suppose $\mu \ge LS(\mathcal{K})$. If $\mathcal{K}$ is $\mu$- and ${\mu }^{+}$-superstable and satisfies ${\mu }^{+}$-symmetry, then for any increasing sequence $\langle M_i\in$\mathcal{K}${\geq\mu^{+}}\mid i<\theta<(\sup|M_i|)^+\rangle$of$\mu^+$-saturatedmodels,$\bigcup{i<\theta}M_i$is$\mu^+$-saturated. We also apply results of VanDieren’s Superstability and Symmetry paper and use towers to transfer symmetry from ${\mu }^{+}$ down to $\mu$ in abstract elementary classes which are both $\mu$- and ${\mu }^{+}$-superstable: Theorem 2: 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.