A criterion for the strong cell decomposition property

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.