Independence over arbitrary sets in NSOP$_1$ theories

TL;DR

This paper investigates Kim-independence over arbitrary sets within NSOP$_1$ theories, establishing key properties like Kim's lemma, symmetry, and the independence theorem under certain conditions.

Contribution

It proves Kim's lemma for Kim-dividing over arbitrary sets in NSOP$_1$ theories assuming forking existence, advancing understanding of independence in model theory.

Findings

Kim's lemma for Kim-dividing over arbitrary sets established

Symmetry of Kim-independence proven

Independence theorem for Lascar strong types demonstrated

Abstract

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP$_{1}$ theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.