Non-forking independence in stable theories

TL;DR

This paper simplifies the understanding of non-forking independence in stable theories by identifying a minimal condition that describes it over models and algebraically closed sets, relying on stationarity of types.

Contribution

It introduces a straightforward condition for non-forking independence in stable theories, extending its applicability without complex assumptions.

Findings

Non-forking independence over models can be characterized by a simple condition.

The description extends to algebraically closed sets under mild assumptions.

Types over models in stable theories are stationary, facilitating the results.

Abstract

We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets, without having to use the full strength of the work of the seminal work of Kim and Pillay. The results in this note (which are surely well-known among most model theorists) essentially use that types over models in a stable theory are stationary.