Quasiminimal abstract elementary classes |
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Authors: | Sebastien Vasey |
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Affiliation: | 1.Department of Mathematical Sciences,Carnegie Mellon University,Pittsburgh,USA |
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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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