Model theory of generic vector space endomorphisms III: Reducts

Abstract

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 $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$. In previous work, we introduced a family $\{T_{\hat{I}}^C¸:C\in \mathcal{C}\}$ of extensions of the theory $T_θ:= T \cup \{\text{``$θ$isanendomorphismof$\mathbb{V}$''}\}$ that parameterizes all consistent extensions of the form $$T_{\hat{I}}¸\cup \left\{{\sum }_k{\bigcap }_l\mathrm{Ker}(\ddot{I}{}_{j,k,l}[\hat{I}¸])={\sum }_k{\bigcap }_l\mathrm{Ker}(\hat{I}{\cdot }_{j,k,l}[\hat{I}¸]):j\in \mathcal{J}\right\},$$ where all sums and intersections are finite, all the $\ddot{I}[\hat{I}¸]$'s and $\hat{I}\cdot [\hat{I}¸]$'s are polynomials over $K$ with $\hat{I}¸$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T_{\hat{I}}^C¸$ has a model companion $T\hat{I}{¸}^C$. We simplify our axiomatization of $T\hat{I}{¸}^C$ 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 $(\mathcal{M},\hat{I}¸)\models T_{\hat{I}}^C¸$ and a polynomial $\ddot{I}\in K[X]$, we show that $(\mathcal{M},\mathrm{Ker}(\ddot{I}[\hat{I}¸]))$ is, unless $\mathrm{Ker}(\ddot{I}[\hat{I}¸])=\{0\}$ or $\mathrm{Ker}(\ddot{I}[\hat{I}¸])=\mathbb{V}$, an existentially closed model of $T_V := T \cup \{\text{``$V$isavectorsubspaceof$\mathbb{V}$''}\}$. In the same vein, we present a criterion for when $(\mathcal{M},\ddot{I}[\hat{I}¸])$ is again an existentially closed model of $T_{\hat{I}}^{C^{\prime }}¸$ for some $C^{\prime }\in \mathcal{C}$.

32 pages