Real Algebraic and Analytic Geometry
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182. Roman Wencel:
Weakly o-minimal non-valuational structures.

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Submission: 2008, January 4.

Abstract:
A weakly o-minimal structure $\mathcal{M}=(M,\leq,+,\ldots)$ expanding an ordered group $(M,\leq,+)$ is called non-valuational iff for every cut $\langle C,D\rangle$ of $(M,\leq)$ definable in $\mathcal{M}$, we have that $\inf\{y-x:x\in C,y\in D\}=0$. The study of non-valuational weakly o-minimal expansions of real closed fields carried out in \cite{MMS} suggests that this class is very close to the class of o-minimal expansions of real closed fields. Here we further develop this analogy. We establish an o-minimal style cell decomposition for weakly o-minimal non-valuational expansions of ordered groups. For structures enjoying such a strong cell decomposition we construct a canonical o-minimal extension. Finally, we make attempts towards generalizing the o-minimal Euler chararacteristic to the class of sets definable in weakly o-minimal structures with strong cell decomposition.

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