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.
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Vasey, S. Quasiminimal abstract elementary classes. Arch. Math. Logic 57, 299–315 (2018). https://doi.org/10.1007/s00153-017-0570-7
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DOI: https://doi.org/10.1007/s00153-017-0570-7
Keywords
- Abstract elementary class
- Quasiminimal pregeometry class
- Pregeometry
- Closure space
- Exchange axiom
- Homogeneity