Model Theory of Generic Vector Space Endomorphisms II

Abstract

This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set $T_θ:= T \cup \{\text{``$θ$ is a $K$-endomorphism of $\mathbb{V}$''}\}$. We previously defined a family $\{T^C_θ: C \in \mathcal{C}\}$ of extensions of $T_θ$ 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, and all the $ρ[θ]$'s and $η[θ]$'s are polynomials over $K$ with $θ$ plugged in. Notice that properties such as $θ^2 - 2\operatorname{Id} = 0$ or ``$ρ[θ]$ is injective for every $ρ\in K[X] \setminus \{0\}$'' can be expressed in such a manner. We also presented a sufficient condition that implies that every $T^C_θ$ has a model companion $Tθ^C$. Under this condition, we characterize all definable sets in $Tθ^C$ and study the completions of $Tθ^C$ as well as the algebraic closure. If $T$ is o-minimal and extends $\operatorname{Th}(\mathbb{R}, <)$, we prove that $Tθ^C$ has an o-minimal open core.