Model theory of generic vector space endomorphisms III: Reducts

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 $\mathrm{\varnothing }$-defines an infinite $K$-vector space $\mathbb{V}$. In previous work, we introduced a family $\{T_{\theta }^C:C\in \mathcal{C}\}$ of extensions of the theory T_θ:= T \cup \{\text{``θ$isanendomorphismof$$\mathbb{V}$''}\} that parameterizes all consistent extensions of the form $$T_{\theta }\cup \{{\sum }_k{\bigcap }_l\mathrm{Ker}({\rho }_{j,k,l}[\theta ])={\sum }_k{\bigcap }_l\mathrm{Ker}({\eta }_{j,k,l}[\theta ]):j\in \mathcal{J}\},$$ where all sums and intersections are finite, all the $\rho [\theta ]$‘s and $\eta [\theta ]$‘s are polynomials over $K$ with $\theta$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T_{\theta }^C$ has a model companion $T{\theta }^C$. We simplify our axiomatization of $T{\theta }^C$ and the criterion for its existence for theories close to the theory of $K$-vector spaces''. We apply this to the explicit case where $T$ is the pure theory of $K$-vector spaces and characterize all $\varnothing$-definable endomorphisms of $\mathbb{V}$ in this case. Given an existentially closed model $(\mathcal{M}, θ) \models T^C_θ$ and a polynomial $ρ\in K[X]$, we show that $(\mathcal{M},\operatorname{Ker}(ρ[θ]))$ is, unless $\operatorname{Ker}(ρ[θ]) = \{0\}$ or $\operatorname{Ker}(ρ[θ]) = \mathbb{V}$, an existentially closed model of $T_V := T \cup \{\text{$V$ is a vector subspace of $\mathbb{V}$’’}}$\mathrm{.}Inthesamevein,wepresentacriterionforwhen$($\mathcal{M}$, ρ[θ])$isagainanexistentiallyclosedmodelof$T^{C’}_θ$forsome$C’ \in $\mathcal{C}$$.