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On some Fraisse limits with free amalgamation

Yvon Bossut

TL;DR

The work develops a general method to produce NSOP$_1$ theories as Fraïssé limits of classes with strong free-amalgamation properties under hypothesis $(H)$. It proves existence and that Kim-independence coincides with algebraic independence, while forking arises from base monotonicity of Kim-forking, and it furnishes a stationary independence relation. The framework is extended via parameterization to obtain broader NSOP$_1$ limits and to characterize when such limits fail to be simple. It also analyzes how adding generic predicates and generic functions preserves $(H)$ and yields explicit non-simple NSOP$_1$ examples. Building on foundational results of $<$Baudisch, Ramsey, Chernikov, Kruckman$>$, the paper situates its constructions within the landscape of NSOP$_1$ model theory and outlines directions for future exploration of stationary-independence NSOP$_1$ theories.

Abstract

In the first part of this work the notion of stable Kim-forking is discussed and some context on this matter is given. In the second part a general way of building some examples of NSOP1 theories as the limit of some Fraisse class satisfying stronger conditions is given. These limits will satisfy existence, that Kim-independence coincide with algebraic independence, and that forking independence is obtained by forcing base monotonicity on Kim-forking over arbitrary sets. These theories also come with a stationary independence relation. This study is based on the results of Baudisch, Ramsey, Chernikov and Kruckman.

On some Fraisse limits with free amalgamation

TL;DR

The work develops a general method to produce NSOP theories as Fraïssé limits of classes with strong free-amalgamation properties under hypothesis . It proves existence and that Kim-independence coincides with algebraic independence, while forking arises from base monotonicity of Kim-forking, and it furnishes a stationary independence relation. The framework is extended via parameterization to obtain broader NSOP limits and to characterize when such limits fail to be simple. It also analyzes how adding generic predicates and generic functions preserves and yields explicit non-simple NSOP examples. Building on foundational results of Baudisch, Ramsey, Chernikov, Kruckman, the paper situates its constructions within the landscape of NSOP model theory and outlines directions for future exploration of stationary-independence NSOP theories.

Abstract

In the first part of this work the notion of stable Kim-forking is discussed and some context on this matter is given. In the second part a general way of building some examples of NSOP1 theories as the limit of some Fraisse class satisfying stronger conditions is given. These limits will satisfy existence, that Kim-independence coincide with algebraic independence, and that forking independence is obtained by forcing base monotonicity on Kim-forking over arbitrary sets. These theories also come with a stationary independence relation. This study is based on the results of Baudisch, Ramsey, Chernikov and Kruckman.
Paper Structure (15 sections, 25 theorems, 28 equations, 9 figures)

This paper contains 15 sections, 25 theorems, 28 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries : Independence relations and NSOP1
  3. Forking and dividing
  4. NSOP1 and the Kim-Pillay theorem
  5. Constructing the limit
  6. Hypothesis on the class and basic examples
  7. Properties of the limit theory
  8. The parameterized class
  9. The definition of the parameterized theory
  10. Basic properties
  11. Amalgamation and Combinatorial Properties
  12. Examples of generic extensions preserving $(H)$
  13. Adding a generic predicate
  14. Adding generic functions
  15. Some perspectives

Key Result

Theorem 2.2.0.2

dobrowolski2022independence A theory $T$ with existence is NSOP$_1$ if and only if Kim-forking satisfies symmetry in a monster model $\mathbb{M}$ of $T$, i.e. $a \mathrel{ \mathop{ \vcenter{ \hbox{\oalign{{}$\vert$\cr {} $\smile$\cr{}}} } }\displaylimits_{} } ^{K}_{e}b$ if and only if $b \mathrel{

Figures (9)

  • Figure 1: The strong order property of the first kind
  • Figure 2: Algebraically Independent 3-Amalgamation
  • Figure 3: Proving Algebraically Independent 3-Amalgamation
  • Figure 4: The embedding we needed
  • Figure 5: The inductive system of $A'\oplus_{E'}B'$
  • ...and 4 more figures

Theorems & Definitions (80)

  • 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
  • Theorem 2.2.0.2
  • Theorem 2.2.0.3
  • Definition 3.1.0.1
  • ...and 70 more