2026/07/21 by Leon Chini
Mathematics · #math.LO
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 \mathbbV. In previous work, we introduced a family \TCθ: C ∈ C\ of extensions of the theory Tθ:= T ∪ \\text``θ is an endomorphism of \mathbbV''\ that parameterizes all consistent extensions of the form Tθ∪ \∑\nolimitsk\bigcap\nolimitslKer(ρj, k, l[θ]) = ∑\nolimitsk\bigcap\nolimitsl Ker(ηj, k, l[θ]) : j ∈ J\, where all sums and intersections are finite, all the ρ[θ]'s and η[θ]'s are polynomials over K with θ plugged in, and J is some possibly infinite index set. We also presented a sufficient condition that implies that every TCθ has a model companion TθC. In this paper, we show that, under this sufficient condition, the model companion TθC has NATP, a neostability property recently introduced by Ahn and Kim, whenever T does.