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

Abstract

We demonstrate the following uniform local definable cell decomposition theorem in this paper. Consider a structure $\mathcal{M}=(M,<,0,+,\dots )$ 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 $M^m\times B=X_1\cup \dots \cup X_k$ such that $B=(X_1{)}_b\cup \dots \cup (X_k{)}_b$ is a definable cell decomposition of $B$ for any $b\in M^m$ and $X_i\cap A_{\lambda }=\varnothing$ or $X_i\subset A_{\lambda }$ for any $1\le i\le k$ and $\lambda \in \Lambda$. Here, the notation $S_b$ denotes the fiber of a definable subset $S$ of $M^{m+n}$ at $b\in M^m$.

Cited by in corpus (2)

• Almost o-minimal structures and $\mathfrak{X}$-structures
• Simple product and locally o-minimal theories