论文标题
DP最少的涂鸦组和整数的估值
Dp-minimal profinite groups and valuations on the integers
论文作者
论文摘要
我们研究了配备有均匀定义的开放亚组基本系统的DP最小无限的小组。我们表明，这些组具有一个开放的子组$ a $，因此，$ a $是一些$ \ mathbb {f} _p $的直接产品，对于某些prime $ p $，或$ a $是$ a \ cong \ prod_p \ prod_p \ prod_p \ prod_p \ mathbb {z} $ _p} $ __p^$ a $ a $ a $ a $ a_p $ a_p每个Prime $ P $的有限ABELIAN $ P $ -Group。此外，我们表明，如果$ a $是此表格，那么就有一个基本的开放子组系统，因此该子组家族的扩展为$ a $是DP-Minimal。我们的主要成分是一类有价值的阿贝尔群体的消除量子结果。我们还将其应用于$（\ mathbb {z}，+）$，我们表明，如果我们通过任何子组$（b_i）_ {i <ω} $扩展$（\ mathbb {z}，+）$，我们获得了DP-Minimal结构。当且仅当商的大小$ b_i/b_ {i+1} $是边界时，此结构是远端的。
We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many copies of $\mathbb{F}_p$ for some prime $p$, or $A$ is of the form $A \cong \prod_p \mathbb{Z}_p^{α_p} \times A_p$ where $α_p < ω$ and $A_p$ is a finite abelian $p$-group for each prime $p$. Moreover, we show that if $A$ is of this form, then there is a fundamental system of open subgroups such that the expansion of $A$ by this family of subgroups is dp-minimal. Our main ingredient is a quantifier elimination result for a class of valued abelian groups. We also apply it to $(\mathbb{Z},+)$ and we show that if we expand $(\mathbb{Z},+)$ by any chain of subgroups $(B_i)_{i<ω}$, we obtain a dp-minimal structure. This structure is distal if and only if the size of the quotients $B_i/B_{i+1}$ is bounded.
