AI 中文总结
研究作用于向量空间自同态的模型伴随，此前引入相关理论扩展族及充分条件，本文证明在此条件下，当\(T\)有NATP时，其模型伴随\(T\theta^C\)也具有NATP。
AI 中文摘要
本文进一步研究作用于向量空间（可能带有额外结构）的自同态的模型伴随。设\(T\)是一个\(\varnothing\)-定义无限\(K\)-向量空间\(\mathbb{V}\)的模型完全理论。在之前工作中，引入了理论\(T_\theta := T \cup \{``\theta\)是\(\mathbb{V}\)的自同态”\}\)的一族扩展\(\{T^C_\theta: C \in \mathcal{C}\}\)，参数化了所有一致扩展形式。还给出了一个充分条件，表明每个\(T^C_\theta\)有模型伴随\(T\theta^C\)。本文证明在该充分条件下，当\(T\)具有NATP（Ahn和Kim最近引入的一种新稳定性性质）时，模型伴随\(T\theta^C\)也具有NATP。
英文摘要
This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Let $T$ be a model-complete theory that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$. In previous work, we introduced a family $\{T^C_θ: C \in \mathcal{C}\}$ of extensions of the theory $T_θ:= T \cup \{\text{``$θ$ is an endomorphism of $\mathbb{V}$''}\}$ that parameterizes all consistent extensions of the form $$ T_θ\cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(ρ_{j, k, l}[θ]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(η_{j, k, l}[θ]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, all the $ρ[θ]$'s and $η[θ]$'s are polynomials over $K$ with $θ$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T^C_θ$ has a model companion $Tθ^C$. In this paper, we show that, under this sufficient condition, the model companion $Tθ^C$ has $\operatorname{NATP}$, a neostability property recently introduced by Ahn and Kim, whenever $T$ does.
Comments40 pages. arXiv admin note: text overlap with arXiv:2502.13667
