Uniform local definable cell decomposition for locally o-minimal expansion of the group of reals

TL;DR

This paper proves a uniform local definable cell decomposition theorem for structures elementarily equivalent to a locally o-minimal expansion of the reals, enabling consistent partitioning of definable sets in such structures.

Contribution

It establishes a uniform local cell decomposition theorem in locally o-minimal structures, extending cell decomposition techniques to a broader class of structures.

Findings

Existence of a finite definable partition compatible with given sets

Uniformity of cell decomposition across fibers

Applicable to structures elementarily equivalent to locally o-minimal expansions

Abstract

We demonstrate the following uniform local definable cell decomposition theorem in this paper. Consider a structure $\mathcal M = (M, <,0,+, \ldots)$ elementarily equivalent to a locally o-minimal expansion of the group of reals $(\mathbb R, <,0,+)$. Let $\{A_\lambda\}_{\lambda\in\Lambda}$ be a finite family of definable subsets of $M^{m+n}$. There exist an open box $B$ in $M^n$ containing the origin and a finite partition of definable sets $M^m \times B = X_1 \cup \ldots \cup X_k$ such that $B=(X_1)_b \cup \ldots \cup (X_k)_b$ is a definable cell decomposition of $B$ for any $b \in M^m$ and $X_i \cap A_\lambda = \emptyset$ or $X_i \subset A_\lambda$ for any $1 \leq i \leq k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b \in M^m$.