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

TL;DR

This paper establishes a structural dividing line for superstable abstract elementary classes based on symmetry properties, without relying on additional set-theoretic assumptions, and explores how symmetry transfers between different cardinalities.

Contribution

It introduces a new dividing line involving $mbda$-symmetry in superstable classes and demonstrates how symmetry at $mbda^+$ implies symmetry at $mbda$ using towers.

Findings

Symmetry at mbda^+ implies symmetry at mbda in superstable classes.

Union of increasing mbda^+-saturated models remains mbda^+-saturated.

The structural dividing line is characterized without extra set-theoretic assumptions.

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 $\mu\geq LS(\mathcal{K})$. 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 $\mu^+$-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.