Quasiminimal abstract elementary classes

TL;DR

This paper introduces quasiminimal abstract elementary classes (AECs), establishing their properties, and connecting them to Zilber's quasiminimal pregeometry classes, showing the exchange axiom is unnecessary.

Contribution

It defines quasiminimal AECs with specific semantic conditions and links them to existing quasiminimal pregeometry classes, expanding the theoretical framework.

Findings

Quasiminimal AECs satisfy four key semantic conditions.

A correspondence between quasiminimal pregeometry classes and AECs is established.

The exchange axiom is shown to be redundant in the definition of quasiminimal pregeometry classes.

Abstract

We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable L\"owenheim-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.