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Non-forking independence in stable theories

Amador Martin-Pizarro

Abstract

We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets, without having to use the full strength of the work of the seminal work of Kim and Pillay. The results in this note (which are surely well-known among most model theorists) essentially use that types over models in a stable theory are stationary.

Non-forking independence in stable theories

Abstract

We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets, without having to use the full strength of the work of the seminal work of Kim and Pillay. The results in this note (which are surely well-known among most model theorists) essentially use that types over models in a stable theory are stationary.

Paper Structure

This paper contains 2 sections, 5 theorems, 24 equations.

Table of Contents

  1. Non-forking and stationarity
  2. Characterizing non-forking in some classical theories

Key Result

Proposition 1.4

Assume that the stable theory $T$ admits a weak notion of independence $\mathop{\ \ \hbox{$\mid^{\hbox{$\mathrm{1}$}}$} \hbox{$\smile$}\ \ }$ which satisfies Stationarity over models. Then the relation $\mathop{\ \ \hbox{$\mid^{\hbox{$\mathrm{1}$}}$} \hbox{$\smile$}\ \ }$ agrees with non-forkin Furthermore, assume the weak notion of independence $\mathop{\ \ \hbox{$\mid^{\hbox{$\mathrm{1}$}}$

Theorems & Definitions (24)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Remark 1.5
  • proof
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • ...and 14 more