Model Theory of Generic Vector Space Endomorphisms IV: Preservation of NATP

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 $\{T_{\theta }^C:C\in \mathcal{C}\}$ of extensions of the theory $T_θ:= T \cup \{\text{``$θ$isanendomorphismof$\mathbb{V}$''}\}$ that parameterizes all consistent extensions of the form $$T_{\theta }\cup \left\{{\sum }_k{\bigcap }_l\mathrm{Ker}({\rho }_{j,k,l}[\theta ])={\sum }_k{\bigcap }_l\mathrm{Ker}({\eta }_{j,k,l}[\theta ]):j\in \mathcal{J}\right\},$$ where all sums and intersections are finite, all the $\rho [\theta ]$'s and $\eta [\theta ]$'s are polynomials over $K$ with $\theta$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T_{\theta }^C$ has a model companion $T{\theta }^C$. In this paper, we show that, under this sufficient condition, the model companion $T{\theta }^C$ has $\mathrm{NATP}$, a neostability property recently introduced by Ahn and Kim, whenever $T$ does.

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