Model Theory of Generic Vector Space Endomorphisms II

This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\mathrm{\varnothing }$-defines an infinite $K$-vector space $\mathbb{V}$ in every model, we set T_θ:= T \cup \{\text{``θ$definesa$K$-endomorphismof$$\mathbb{V}$"}\}. We previously defined a family $\{T_{\theta }^C:C\in \mathcal{C}\}$ of extensions of $T_{\theta }$ which parameterizes all consistent extensions of the form $$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, and all the $\rho [\theta ]$'s and $\eta [\theta ]$'s are polynomials over $K$ with $\theta$ plugged in. Notice that properties such as ${\theta }^2-2\mathrm{Id}=0$ 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 $T_{\theta }^C$ 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 $\mathrm{Th}(\mathbb{R},<)$, we prove that $T{\theta }^C$ has o-minimal open core.