Model theory of generic vector space endomorphisms III: Reducts

Abstract

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 $C' \in \mathcal{C}$.