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

Abstract

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 $⟨M_i\in {\mathcal{K}}_{\ge {\mu }^{+}}\mid i<\theta <(\sup \Vert M_i\Vert {)}^{+}⟩$ of ${\mu }^{+}$-saturated models, ${\bigcup }_{i<\theta }{M}_i$ 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.

This paper is a synthesis of arXiv:1507.01991 and arXiv:1507.01989

References in corpus (6)

• Building independence relations in abstract elementary classes
• Downward categoricity from a successor inside a good frame
• Building prime models in fully good abstract elementary classes
• Transferring Symmetry
• On the structure of categorical abstract elementary classes with amalgamation
• A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes

Cited by in corpus (9)

• Saturation and solvability in abstract elementary classes with amalgamation
• Downward categoricity from a successor inside a good frame
• Toward a stability theory of tame abstract elementary classes
• Symmetry in abstract elementary classes with amalgamation
• Building prime models in fully good abstract elementary classes
• Abstract elementary classes stable in ${\aleph }_0$
• A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes
• Tameness from two successive good frames
• Superstability and Symmetry