A criterion for the strong cell decomposition property

Let M = (M, <, ...) be a weakly o-minimal structure. Assume that Def (M) is the collection of all definable sets of M and for any m is an element of N, Def(m)(M) is the collection of all definable subsets of M-m in M. We show that the structure M has the strong cell decomposition property if and only if there is an o-minimal structure N such that Def (M) = {Y & cap; M-m : m is an element of N, Y is an element of Def(m)(N)}. Using this result, we prove that: (a) Every induced structure has the strong cell decomposition property. (b) The structure M has the strong cell decomposition property if and only if the weakly o-minimal structure M-M(& lowast;) 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.

Automated summary

Assuming M = (M, <, ...) is a weakly o-minimal structure, if there exists an o-minimal structure N such that Def(m)(M) is the collection of all definable subsets of M-m in M for any m in N, then the structure M has the strong cell decomposition property.