Quasiminimal abstract elementary classes

Article

Abstract

We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim–Skolem–Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber’s quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber’s definition of a quasiminimal pregeometry class.

Keywords

Abstract elementary class Quasiminimal pregeometry class Pregeometry Closure space Exchange axiom Homogeneity 

Mathematics Subject Classification

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

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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