We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.

中文翻译：

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稠密集的 o 极小结构驯服扩展中的结构定理

我们研究可在 o 最小结构的驯服扩展中定义的集合和组。令 $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ 是一个 o 极小 $\mathcal L$-结构 $\cal M$ 的稠密集 $P$ 的扩展，使得三个驯服条件成立。我们证明了可定义集合和函数的结构定理，类似于已知的 o 最小结构的有影响的单元分解定理。结构定理在 $\mathcal {\widetilde M}$ 的所有已知例子中推进了最先进的技术，因为它实现了将可定义集合分解为 `cones' 的 \emph{unions}，而不仅仅是布尔值它们的组合。我们还在温和的环境中发展了正确的维度理论。应用包括：(i) 可定义集的维数与合适的预几何维数一致，