Mann对中的小集合

摘要：

设$\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$是实封闭的域$\mathcal M$通过具有Mann性质的$\langle M^{>0}, \cdot\rangle$的稠密子群$G$的展开.我们证明了由$\mathcal M$在$G$上诱导的结构消除了不确定性。因此，在$\mathcal M$中可定义的每个小集合$X$可以在参数上均匀地可定义地嵌入到某些$G^l$中。这些结果证明在一个更一般的设置，其中$\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$是一个O-最小结构$\mathcal M$的密集集$P\subseteq M$，满足三个驯服条件的扩展。

英文摘要：

Let $\widetilde{\mathcal M}=\langle \mathcal M, G\rangle$ be an expansion of a real closed field $\mathcal M$ by a dense subgroup $G$ of $\langle M^{>0}, \cdot\rangle$ with the Mann property. We prove that the induced structure on $G$ by $\mathcal M$ eliminates imaginaries. As a consequence, every small set $X$ definable in $\mathcal M$ can be definably embedded into some $G^l$, uniformly in parameters. These results are proved in a more general setting, where $\widetilde{\mathcal M}=\langle \mathcal M, P\rangle$ is an expansion of an o-minimal structure $\mathcal M$ by a dense set $P\subseteq M$, satisfying three tameness conditions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。