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Open cell property in weakly o-minimal structures

Tomohiro Kawakami, Hiroshi Tanaka

TL;DR

The paper addresses extending Wilkie's open-cell decomposition, known in o-minimal structures, to weakly o-minimal settings, including expansions of real closed fields and ordered groups. It develops strong and refined strong cells, their completions $\overline{C}$, and the canonical o-minimal extension $\overline{\mathcal{M}}$, establishing weak definable choice and transferring cell-decomposition ideas to the weak framework. A key contribution is the characterization of non-valuational weakly o-minimal structures via strong cell decomposition, plus showing that definable open sets in semi-bounded, non-valuational ordered groups decompose into finitely many open strong cells. The results extend tame geometric phenomena to a broader class of structures, providing structured descriptions of open definable sets and bridging weakly o-minimal and o-minimal theories via completions.

Abstract

In an o-minimal structure, Wilkie proved that every bounded definable open set is a finite union of definable open cells. We generalze it to weakly o-minimal structures.

Open cell property in weakly o-minimal structures

TL;DR

The paper addresses extending Wilkie's open-cell decomposition, known in o-minimal structures, to weakly o-minimal settings, including expansions of real closed fields and ordered groups. It develops strong and refined strong cells, their completions , and the canonical o-minimal extension , establishing weak definable choice and transferring cell-decomposition ideas to the weak framework. A key contribution is the characterization of non-valuational weakly o-minimal structures via strong cell decomposition, plus showing that definable open sets in semi-bounded, non-valuational ordered groups decompose into finitely many open strong cells. The results extend tame geometric phenomena to a broader class of structures, providing structured descriptions of open definable sets and bridging weakly o-minimal and o-minimal theories via completions.

Abstract

In an o-minimal structure, Wilkie proved that every bounded definable open set is a finite union of definable open cells. We generalze it to weakly o-minimal structures.

Paper Structure

This paper contains 3 sections, 6 theorems.

Table of Contents

  1. Introduction
  2. Weakly o-minimal expansions of real closed fields
  3. Weakly o-minimal expansions of ordered groups

Key Result

Proposition 2.8

Suppose that $\mathcal{M} = (M, <, +, \ldots)$ is a weakly o-minimal expansion of an ordered group $(M, <, +)$. Let $S \subseteq M^{m+n}$ be definable and $\pi : M^{m+n} \to M^m$ the projection of the first $m$ coordinates. Then, there exists a definable map $f : \pi(S) \to \overline{M}^n$ such that

Theorems & Definitions (16)

  • Definition 2.1: W1
  • Definition 2.2
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8: Weak definable choice
  • proof
  • Proposition 2.9
  • proof
  • Example 2.10
  • ...and 6 more