Kim-forking for hyperimaginaries in NSOP1 theories

TL;DR

This paper extends Kim-independence properties to hyperimaginaries in NSOP1 theories, establishing key independence results and applying them to types and bases, advancing model theory understanding.

Contribution

It introduces the adaptation of Kim-independence to hyperimaginaries in NSOP1 theories, proving core properties and deriving new results on types and bases.

Findings

Kim-independence over hyperimaginaries satisfies key properties

Results on Lascar strong types and Kim-forking using boundedly closed hyperimaginaries

Extension of Kim's colinearity and canonical bases to hyperimaginaries

Abstract

We adapt the properties of Kim-independence in NSOP1 theories with existence proven in [5],[4] and [2] by Ramsey, Kaplan, Chernikov, Dobrowolski and Kim to hyperimaginaries by adding the assumption of existence for hyperimaginaries. We show that Kim-independence over hyperimaginaries satisfies a version of Kim's lemma, symmetry, the independence theorem, transitivity and witnessing. As applications we adapt Kim's results around colinearity and weak canonical bases from [8] to hyperimaginaries and give some new results about Lascar strong types and Kim-forking using boundedly closed hyperimaginaries.