Symmetry in abstract elementary classes with amalgamation

TL;DR

This paper investigates the symmetry property in abstract elementary classes with amalgamation, establishing transfer results, connections with tameness, and implications for categoricity and structural properties.

Contribution

It introduces new transfer results for symmetry in AECs, links symmetry with tameness and superstability, and extends classical categoricity results to broader contexts.

Findings

Downward transfer of symmetry in AECs with amalgamation.

Symmetry can be deduced from failure of the order property.

Superstability and tameness together imply symmetry.

Abstract

This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.