A criterion for the strong cell decomposition property

Abstract

Let ${\mathcal {M}}=(M, <, \ldots )$ be a weakly o-minimal structure. Assume that ${\mathcal {D}}ef({\mathcal {M}})$ is the collection of all definable sets of ${\mathcal {M}}$ and for any $m\in {\mathbb {N}}$, ${\mathcal {D}}ef_m({\mathcal {M}})$ is the collection of all definable subsets of $M^m$ in ${\mathcal {M}}$. We show that the structure ${\mathcal {M}}$ has the strong cell decomposition property if and only if there is an o-minimal structure ${\mathcal {N}}$ such that ${\mathcal {D}}ef({\mathcal {M}})=\{Y\cap M^m: \ m\in {\mathbb {N}}, Y\in {\mathcal {D}}ef_m({\mathcal {N}})\}$. Using this result, we prove that: (a) Every induced structure has the strong cell decomposition property. (b) The structure ${\mathcal {M}}$ has the strong cell decomposition property if and only if the weakly o-minimal structure ${\mathcal {M}}^*_M$ has the strong cell decomposition property. Also we examine some properties of non-valuational weakly o-minimal structures in the context of weakly o-minimal structures admitting the strong cell decomposition property.