Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
Our main result (Theorem 1) suggests a possible dividing line ($μ$-superstable $+$ $μ$-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 $μ^+$ which satisfies the joint embedding and amalgamation properties. Suppose $μ\geq LS(\mathcal{K})$. If $\mathcal{K}$ is $μ$- and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\in\mathcal{K}_{\geqμ^{+}}\mid i<θ<(\sup\|M_i\|)^+\rangle$ of $μ^+$-saturated models, $\bigcup_{i<θ}M_i$ is $μ^+$-saturated. We also apply results of VanDieren's Superstability and Symmetry paper and use towers to transfer symmetry from $μ^+$ down to $μ$ in abstract elementary classes which are both $μ$- and $μ^+$-superstable: Theorem 2: Suppose $\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\mathcal{K}$ is both $μ$- and $μ^+$-superstable. If $\mathcal{K}$ has symmetry for non-$μ^+$-splitting, then $\mathcal{K}$ has symmetry for non-$μ$-splitting.