论文标题
丰富整数的谓词和驯服的扩展
Enriching a predicate and tame expansions of the integers
论文作者
论文摘要
给定一个结构$ \ MATHCAL {M} $和一个稳定的嵌入式$ \ emptyset $ -definable set $ q $，当通过某种进一步的结构$ \ Mathcal {q} $上富含$ q $的诱导结构时，我们就证明了可驯服的保存结果。特别是，我们表明，如果$ t = \ text {th}（\ Mathcal {m}）$和$ \ text {th}（\ Mathcal {q}）$是稳定的（superstable，$ω$ - 稳定），那么理论$ t [\ nathcal of the Mathcal {Q} $ {m Mathcal of the Mathcal of the Mathcal {m Mathcal {Q} $ $ \ MATHCAL {Q} $。假设$ t $的简单性，消除了超级磁性以及与代数关闭行为有关的$ q $的进一步条件，我们还表明，简单性和NSOP $ _1 $通过$ \ text {th}（\ nathcal {q}）$ to $ \ text {t text {t}（\ natcal {q}）$ to $ t to $ t to $ t [$ t [\ mathcal {q} Q} $。然后，我们证明了弱化结构，尤其是整数组的驯服扩展的几种应用。例如，我们构建了$（\ mathbb {z}，+）$的严格稳定扩展的第一个已知示例。更一般地，我们表明，可以在稳定的（spec。，Suppertable，Sipper，Nip，NTP $ _2 $，nsop $ _1 $）扩展$（\ ntp $ _1 $（Z）$（Z Z} Z} Z} Z}， $ a \ subseteq \ mathbb {n} $。
Given a structure $\mathcal{M}$ and a stably embedded $\emptyset$-definable set $Q$, we prove tameness preservation results when enriching the induced structure on $Q$ by some further structure $\mathcal{Q}$. In particular, we show that if $T=\text{Th}(\mathcal{M})$ and $\text{Th}(\mathcal{Q})$ are stable (resp., superstable, $ω$-stable), then so is the theory $T[\mathcal{Q}]$ of the enrichment of $\mathcal{M}$ by $\mathcal{Q}$. Assuming simplicity of $T$, elimination of hyperimaginaries and a further condition on $Q$ related to the behavior of algebraic closure, we also show that simplicity and NSOP$_1$ pass from $\text{Th}(\mathcal{Q})$ to $T[\mathcal{Q}]$. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of $(\mathbb{Z},+)$. More generally, we show that any stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) countable graph can be defined in a stable (resp., superstable, simple, NIP, NTP$_2$, NSOP$_1$) expansion of $(\mathbb{Z},+)$ by some unary predicate $A\subseteq\mathbb{N}$.
