Independence relations for exponential fields

TL;DR

This paper introduces four canonical independence relations in exponential fields, analyzing their properties and stability, including NSOP$_1$-like, non-simple, and stable behaviors, with a focus on Zilber's exponential fields.

Contribution

It defines and characterizes four distinct independence relations in exponential fields, connecting them to model-theoretic properties like stability and simplicity.

Findings

Two relations are NSOP$_1$-like and non-simple

One relation is stable

The quasiminimal pregeometry is stable and uncountably categorical

Abstract

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP$_1$-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.