Symmetry in abstract elementary classes with amalgamation

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.

37 pages. This merges with arXiv:1509.01488 . Was previously titled "Transferring symmetry downward and applications"

References in corpus (3)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes

Cited by in corpus (7)

• Saturation and solvability in abstract elementary classes with amalgamation
• Toward a stability theory of tame abstract elementary classes
• Abstract elementary classes stable in ${\aleph }_0$
• Building prime models in fully good abstract elementary classes
• Good Frames in the Hart-Shelah Example
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Tameness from two successive good frames