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Stable formulas in ordered structures

Daniel Max Hoffmann, Chieu-Minh Tran, Jinhe Ye

Abstract

We classify the stable formulas in the theory of Dense Linear Orders without endpoints, the stable formulas in the theory of Divisible Abelian Groups, and the stable formulas without parameters in the theory of Real Closed Fields. The third result, unexpectedly, requires the Hironaka's theorem on resolution of singularities.

Stable formulas in ordered structures

Abstract

We classify the stable formulas in the theory of Dense Linear Orders without endpoints, the stable formulas in the theory of Divisible Abelian Groups, and the stable formulas without parameters in the theory of Real Closed Fields. The third result, unexpectedly, requires the Hironaka's theorem on resolution of singularities.

Paper Structure

This paper contains 5 sections, 12 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Large stable sets
  4. Stable formulas in DLO and DOAG
  5. Stable formulas in RCF

Key Result

Proposition 1.2

Suppose $\varphi(x;y)$ is stable, and $\dim \varphi(\mathscr{M}) = |x|+|y|$ . Then there is an $L(M)$-formula $\varphi'(x;y)$ which is a disjuntion of rectangular $L(M)$-formulas such that $\dim( \varphi(\mathscr{M}) \triangle \varphi'(\mathscr{M})) < |x|+|y|$.

Theorems & Definitions (26)

  • Example 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Example 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 16 more