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| Paper IPM / M / 17414 |
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| Abstract: | |
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Let $ \mathcal{M}=(M, <, \cdots) $ 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.
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