Forking, Imaginaries and other features of ACFG

TL;DR

This paper explores the model-theoretic properties of ACFG, an algebraically closed field with an additive subgroup predicate, focusing on imaginaries and independence relations in this non-simple, NSOP1 theory.

Contribution

It provides a detailed analysis of imaginaries and independence relations in ACFG, expanding understanding of its model-theoretic structure.

Findings

Characterization of imaginaries in ACFG

Analysis of Kim-independence and forking independence interactions

Identification of ACFG as an NSOP1 non-simple theory

Abstract

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm{ACFG}$. This theory was introduced recently as a new example of $\mathrm{NSOP}_1$ non simple theory. In this paper we describe more features of $\mathrm{ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm{ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.