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Weakly minimal groups with a new predicate
Journal of Mathematical Logic ( IF 0.9 ) Pub Date : 2019-12-02 , DOI: 10.1142/s0219061320500117
Gabriel Conant 1 , Michael C. Laskowski 2
Affiliation  

Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text].

中文翻译:

具有新谓词的弱最小群

修复弱最小(即超稳定[公式:见文本]-秩[公式:见文本])结构[公式:见文本]。令 [公式:见文本] 是基本子结构的常数展开,让 [公式:见文本] 是宇宙 [公式:见文本] 的任意子集。我们证明展开式 [Formula: see text] 中的所有公式都等价于有界公式,因此 [Formula: see text] 是稳定的(或 NIP)当且仅当 [Formula: see text] 诱导结构 [Formula :[公式:见文本]上的文本]是稳定的(或NIP)。然后我们限制[Formula: see text] 是一个具有弱极小理论的纯阿贝尔群,并且[Formula: see text] 是互代数的(等效地,弱极小与平凡的分叉)。这个设置包含了最近关于 [公式:见文本] 的稳定扩展的大部分研究。使用互代数的各种表征,我们给出了[公式:见正文]形式的稳定结构的新例子。最值得注意的是,我们证明如果 [公式:见文本] 是代数数的弱最小加法子群,则 [公式:见文本] 由具有代数系数的齐次线性递推关系枚举,并且没有特征多项式的重复根[公式:见正文] 是统一的根,那么 [公式:见正文] 对任何 [公式:见正文] 都是超稳定的。
更新日期:2019-12-02
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