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 $T_\theta := T \cup \{\text{``$\theta$ defines a $K$-endomorphism of $\mathbb{V}$"}\}$. We previously defined a family $\{T^C_\theta : C \in \mathcal{C}\}$ of extensions of $T_\theta$ which parameterizes all consistent extensions of the form $$ T_\theta \cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j, k, l}[\theta]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(\eta_{j, k, l}[\theta]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, and all the $\rho[\theta]$'s and $\eta[\theta]$'s are polynomials over $K$ with $\theta$ plugged in. Notice that properties such as $\theta^2 - 2\operatorname{Id} = 0$ or ``$\rho[\theta]$ is injective for every $\rho \in K[X] \setminus \{0\}$" can be expressed in such a manner. We also presented a sufficient condition which implies that every $T^C_\theta$ has a model companion $T\theta^C$. Under this condition, we characterize all definable sets in $T\theta^C$ and use this to study the completions of $T\theta^C$, as well as the algebraic closure. If $T$ is o-minimal and extends $\operatorname{Th}(\mathbb{R}, <)$, we prove that $T\theta^C$ has o-minimal open core.