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A note on some example of NSOP1 theories

Yvon Bossut

TL;DR

This work surveys NSOP$_1$ theories with emphasis on Kim-forking and base-monotonicity phenomena, introducing and analyzing a two-sorted bilinear-form theory $sT^{K}_{ ty}$ built on a NSOP$_1$ field $T_K$. It shows that Kim-independence in this bilinear context often reduces to a field-theoretic notion $K$-independence over models (under suitable existence assumptions) and that several independence properties transfer from the field sort to the two-sorted structure, including an independence theorem. The paper also provides concrete counterexamples demonstrating that forcing base monotonicity does not guarantee extension in NSOP$_1$ theories, first on home sorts within $sT^{ACF}_{ ty}$ and then in the imaginary expansion $sT^{ACF,eq}_{ ty}$, illustrating that $ ext{Ind}^{K}$ and $ ext{Ind}^{f}$ can diverge in these settings. Collectively, these results underscore the delicate and theory-specific relationship between Kim-independence and forking in NSOP$_1$ contexts, with implications for imaginaries and the broader landscape of non-simple NSOP$_1$ theories.

Abstract

We present here some known and some new examples of non-simple NSOP1 theories and some behaviour that Kim-forking can exhibit in these theories, in particular that Kim-forking after forcing base monotonicity can or can not satisfy extension (on arbitrary sets). This study is based on the results of Chernikov, Ramsey, Dobrowolski and Granger.

A note on some example of NSOP1 theories

TL;DR

This work surveys NSOP theories with emphasis on Kim-forking and base-monotonicity phenomena, introducing and analyzing a two-sorted bilinear-form theory built on a NSOP field . It shows that Kim-independence in this bilinear context often reduces to a field-theoretic notion -independence over models (under suitable existence assumptions) and that several independence properties transfer from the field sort to the two-sorted structure, including an independence theorem. The paper also provides concrete counterexamples demonstrating that forcing base monotonicity does not guarantee extension in NSOP theories, first on home sorts within and then in the imaginary expansion , illustrating that and can diverge in these settings. Collectively, these results underscore the delicate and theory-specific relationship between Kim-independence and forking in NSOP contexts, with implications for imaginaries and the broader landscape of non-simple NSOP theories.

Abstract

We present here some known and some new examples of non-simple NSOP1 theories and some behaviour that Kim-forking can exhibit in these theories, in particular that Kim-forking after forcing base monotonicity can or can not satisfy extension (on arbitrary sets). This study is based on the results of Chernikov, Ramsey, Dobrowolski and Granger.
Paper Structure (8 sections, 15 theorems, 6 equations)

This paper contains 8 sections, 15 theorems, 6 equations.

Table of Contents

  1. Context and notations :
  2. Generic bilinear form on an NSOP$_1$ field
  3. NSOP$_1$ and Kim-independence
  4. More properties of independence relations
  5. Proving existence in the case of simple fields
  6. Forcing Base Monotonicity of Kim-Independence
  7. A counter-example for the home sorts
  8. A counter-example for $sT^{ACF,eq}_{\infty}$

Key Result

Lemma 2.1.0.1

Let $(\lambda_{i})_{i<n}$ be a tuple of scalars, $(b_{i})_{i<n}$ be linearly independent vectors in $\mathbb{M}$, $\alpha \in \mathbb{K}$ and $\overline{e}$ be a finite tuple of vectors. Then there is $x\in \mathbb{V}$ such that $[x,x]=\alpha$, $[x,b_{i}]=\lambda_{i}$ for all $i<n$ and $x \centernot

Theorems & Definitions (35)

  • Definition 1.0.0.1
  • Definition 1.0.0.2
  • Lemma 2.1.0.1
  • proof
  • Corollary 2.1.0.2
  • proof
  • Definition 2.1.0.3
  • Proposition 2.1.0.4
  • proof
  • Proposition 2.2.0.1
  • ...and 25 more