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.