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A note on stable Kim-forking

Yvon Bossut

TL;DR

This note introduces weak stable Kim-forking as an NSOP$_1$-theory analogue of stable forking and studies its consequences. It proves that stable Kim-forking over models forces simplicity, while weak stable Kim-forking over models yields NSOP$_1$ without guaranteeing simplicity, clarifying where stability-based arguments can be carried in NSOP$_1$ theories. Concrete examples, such as $T_{eq,P}$ and infinite-dimensional vector spaces with a generic bilinear form, illustrate how stable formulas can witness Kim-forking in practice. The work also develops definability and canonical-base phenomena over algebraically closed sets in the context of $T^{eq}$ and explores the challenges of passing these properties to imaginaries, ending with several open questions for future research. Overall, the paper broadens the transfer of stable-forking techniques to NSOP$_1$ settings and maps out the boundaries and questions surrounding imaginaries and $T^{eq}$.

Abstract

We define weak stable Kim-forking, a notion that generalizes stable forking to the context of NSOP1 theories. We adapt some of the known results on stable forking to this context.

A note on stable Kim-forking

TL;DR

This note introduces weak stable Kim-forking as an NSOP-theory analogue of stable forking and studies its consequences. It proves that stable Kim-forking over models forces simplicity, while weak stable Kim-forking over models yields NSOP without guaranteeing simplicity, clarifying where stability-based arguments can be carried in NSOP theories. Concrete examples, such as and infinite-dimensional vector spaces with a generic bilinear form, illustrate how stable formulas can witness Kim-forking in practice. The work also develops definability and canonical-base phenomena over algebraically closed sets in the context of and explores the challenges of passing these properties to imaginaries, ending with several open questions for future research. Overall, the paper broadens the transfer of stable-forking techniques to NSOP settings and maps out the boundaries and questions surrounding imaginaries and .

Abstract

We define weak stable Kim-forking, a notion that generalizes stable forking to the context of NSOP1 theories. We adapt some of the known results on stable forking to this context.

Paper Structure

This paper contains 10 sections, 13 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries : Independence relations and local types
  3. Without the existence axiom
  4. Assuming existence
  5. Local types
  6. Weak stable Kim-forking over models
  7. Over algebraically closed sets
  8. Examples of theories with weak stable Kim-forking:
  9. The question of imaginaries
  10. Some questions

Key Result

Lemma 3.0.0.3

Let $M\subseteq N$ be models of a theory $T$ with weak stable forking over models. For any type $p=\hbox{\rm tp}(a/N)$, $a \mathrel{ \mathop{ \vcenter{ \hbox{\oalign{{}$\vert$\cr {} $\smile$\cr{}}} } }\displaylimits_{} } ^{f}_{M}N$ if and only if for every stable formula $\varphi(x,y,m)$ with $m\in

Theorems & Definitions (48)

  • Definition 2.1.0.1
  • Definition 2.1.0.2
  • Remark 2.1.0.3
  • Definition 2.1.0.4
  • Definition 2.1.0.5
  • Definition 2.1.0.6
  • Definition 2.2.0.1
  • Definition 2.2.0.2
  • Definition 2.3.0.1
  • Definition 2.3.0.2
  • ...and 38 more