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arXiv · 1611.07380

Quasiminimal abstract elementary classes

Sebastien Vasey
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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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Sebastien Vasey. 2017-06-12. Quasiminimal abstract elementary classes. https://doi.org/10.1007/s00153-017-0570-7

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