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$isa$K$-endomorphismof$\mathbb{V}$''}\}$. We previously defined a family $\{T_{\hat{I}}^C¸:C\in \mathcal{C}\}$ of extensions of $T_{\hat{I}}¸$ that parameterizes all consistent extensions of the form $$T_{\hat{I}}¸\cup \left\{{\sum }_k{\bigcap }_l\mathrm{Ker}(\ddot{I}{}_{j,k,l}[\hat{I}¸])={\sum }_k{\bigcap }_l\mathrm{Ker}(\hat{I}{\cdot }_{j,k,l}[\hat{I}¸]):j\in \mathcal{J}\right\},$$ where all sums and intersections are finite, and all the $\ddot{I}[\hat{I}¸]$'s and $\hat{I}\cdot [\hat{I}¸]$'s are polynomials over $K$ with $\hat{I}¸$ plugged in. Notice that properties such as $\hat{I}{¸}^2-2\mathrm{Id}=0$ or ``$\ddot{I}[\hat{I}¸]$ is injective for every $\ddot{I}\in K[X]\setminus \{0\}$'' can be expressed in such a manner. We also presented a sufficient condition that implies that every $T_{\hat{I}}^C¸$ has a model companion $T\hat{I}{¸}^C$. Under this condition, we characterize all definable sets in $T\hat{I}{¸}^C$ and study the completions of $T\hat{I}{¸}^C$ as well as the algebraic closure. If $T$ is o-minimal and extends $\mathrm{Th}(\mathbb{R},<)$, we prove that $T\hat{I}{¸}^C$ has an o-minimal open core.

28 pages. arXiv admin note: text overlap with arXiv:2502.13667