DP-极小投射有限群与整数赋值
Dp-minimal profinite groups and valuations on the integers
摘要：
研究了具有一致可定义的基本开子群系的dp-极小无限投射有限群。我们证明了这些群有一个开子群$A$，使得对于某个素数$p$，$A$是$\mathbb{F}_p$的可数多个拷贝的直积，或者$A$是$A \cong \prod_p \mathbb{Z}_p^{α_p} \times A_p$的形式，其中对于每个素数$p$，$α_p < ω$和$A_p$是有限阿贝尔$p$ -群.此外，我们证明，如果$A$是这种形式，那么有一个基本系统的开子群，使$A$由这个家庭的子群是dp-最小的。我们的主要内容是一类值交换群的量词消去结果。我们也将其应用到$(\mathbb{Z},+)$，我们表明，如果我们扩大$(\mathbb{Z},+)$由任何链的子群$(B_i)_{i<ω}$，我们得到一个dp-最小结构。这个结构是远端的，当且仅当子$B_i/B_{i+1}$的大小是有界的。
英文摘要：
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.
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