On the uniqueness property of forking in abstract elementary classes

TL;DR

This paper proves the uniqueness of a specific independence notion called μ-forking in abstract elementary classes under superstability conditions, resolving longstanding technical issues and unifying different forking symmetry concepts.

Contribution

It establishes the uniqueness property of μ-forking in abstract elementary classes, bridging two existing notions of forking symmetry and advancing the theory of independence in this context.

Findings

Proves the uniqueness property of μ-forking in superstable AECs.

Shows the equivalence of two different forking symmetry definitions.

Addresses a longstanding technical difficulty in constructing forking notions.

Abstract

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $\mu$-forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking-like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.