Model Theory of Generic Vector Space Endomorphisms II

TL;DR

This paper investigates the model theory of endomorphisms on vector spaces, characterizing extensions and conditions for model companions, with applications to o-minimal theories like real closed fields.

Contribution

It extends previous work by providing a detailed analysis of model companions for endomorphisms with additional structure, including definable sets and algebraic closure.

Findings

Characterization of all consistent extensions of endomorphism theories

Conditions under which model companions exist and are well-behaved

Proof that certain theories have o-minimal open core

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.