Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 423–452 | Cite as

Symmetry in abstract elementary classes with amalgamation

Article

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.

Keywords

Abstract elementary classes Categoricity Superstability Tameness Symmetry Splitting Good frame Limit models Saturated models 

Mathematics Subject Classification

Primary 03C48 Secondary 03C45 03C52 03C55 03C75 03E55 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsRobert Morris UniversityMoon TownshipUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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