论文标题
$（\ Mathbb {z}，+）$的自动机和驯服扩展
Automata and tame expansions of $(\mathbb{Z},+)$
论文作者
论文摘要
在这里解决了哪些自动整数稳定的表征问题。给定一个正整数$ d $和一个子集$ a \ subseteq \ subseteq \ mathbb {z} $，其一组表示$ d $由有限的自动机识别，在$ x+y \中发现了必要的条件，以在$中成为$ x+y \，以在$ \ \ perperatornAme {t th}（\ mathbbbbbbbb {\ sathbb {z} $）中，在$ \ \ perperatornAmeName中稳定。结合Moosa和Scanlon的定理，这给出了$ d $ - 自动$ a \ subseteq \ mathbb {z} $的组合表征，使得$（\ mathbb {z}，+，a）$是稳定的。这种表征是根据Moosa和Scanlon的“ $ F $ - 集”以及Derksen的“基本$ p $ nested sets”。自动机理论方法还用于产生$（\ mathbb {z}，+）$的一些NIP扩展，尤其是monoid $（d^\ mathbb {n}，\ times）$的扩展。
The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$ whose set of representations base $d$ is recognized by a finite automaton, a necessary condition is found for $x+y\in A$ to be a stable formula in $\operatorname{Th}(\mathbb{Z},+,A)$. Combined with a theorem of Moosa and Scanlon this gives a combinatorial characterization of the $d$-automatic $A\subseteq \mathbb{Z}$ such that $(\mathbb{Z},+,A)$ is stable. This characterization is in terms of what were called "$F$-sets" by Moosa and Scanlon and "elementary $p$-nested sets" by Derksen. Automata-theoretic methods are also used to produce some NIP expansions of $(\mathbb{Z},+)$, in particular the expansion by the monoid $(d^\mathbb{N},\times )$.
