AI 中文总结
研究向量空间自同态的模型伴随，简化其扩张理论\(T^C_\theta\)的公理化及存在性准则，应用于\(K -\)向量空间纯理论情形，刻画可定义自同态，并给出相关存在封闭模型的判定准则。
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\(\mathbb{V}\)的模型完备理论。在之前工作中，引入了理论\(T_\theta := T\cup\{\text{“\(\theta\)是\(\mathbb{V}\)的自同态”}\}\)的一族扩张\(\{T^C_\theta : C\in\mathcal{C}\}\)，它参数化了所有形如\(T_\theta\cup\left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j,k,l}[\theta])=\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\eta_{j,k,l}[\theta]) : j\in\mathcal{J}\right\}\)的一致扩张，其中所有和与交都是有限的，所有\(\rho[\theta]\)和\(\eta[\theta]\)都是\(K\)上带\(\theta\)的多项式，\(\mathcal{J}\)是某个可能无限的指标集。还给出了一个充分条件，意味着每个\(T^C_\theta\)都有模型伴随\(T\theta^C\)。简化了\(T\theta^C\)的公理化及其存在性的准则，用于“接近\(K -\)向量空间理论”的理论。应用到\(T\)是\(K -\)向量空间的纯理论的显式情形，刻画了此时\(\mathbb{V}\)的所有\(\varnothing -\)可定义自同态。给定一个存在封闭模型\((\mathcal{M},\theta)\models T^C_\theta\)和多项式\(\rho\in K[X]\)，证明了除非\(\operatorname{Ker}(\rho[\theta]) = \{0\}\)或\(\operatorname{Ker}(\rho[\theta])=\mathbb{V}\)，\((\mathcal{M},\operatorname{Ker}(\rho[\theta]))\)是\(T_V := T\cup\{\text{“\(V\)是\(\mathbb{V}\)的向量子空间”}\}\)的存在封闭模型。同样，给出了\((\mathcal{M},\rho[\theta])\)何时再次是某个\(C'\in\mathcal{C}\)的\(T^{C'}_\theta\)的存在封闭模型的准则。
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 $Tθ^C$. We simplify our axiomatization of $Tθ^C$ and the criterion for its existence for theories ``close to the theory of $K$-vector spaces''. We apply this to the explicit case where $T$ is the pure theory of $K$-vector spaces and characterize all $\varnothing$-definable endomorphisms of $\mathbb{V}$ in this case. Given an existentially closed model $(\mathcal{M}, θ) \models T^C_θ$ and a polynomial $ρ\in K[X]$, we show that $(\mathcal{M},\operatorname{Ker}(ρ[θ]))$ is, unless $\operatorname{Ker}(ρ[θ]) = \{0\}$ or $\operatorname{Ker}(ρ[θ]) = \mathbb{V}$, an existentially closed model of $T_V := T \cup \{\text{``$V$ is a vector subspace of $\mathbb{V}$''}\}$. In the same vein, we present a criterion for when $(\mathcal{M}, ρ[θ])$ is again an existentially closed model of $T^{C'}_θ$ for some $C' \in \mathcal{C}$.
Comments32 pages
