2026/07/19 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. We simplify our axiomatization of TθC 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 \mathbbV in this case. Given an existentially closed model (M, θ) \models TCθ and a polynomial ρ∈ K[X], we show that (M,Ker(ρ[θ])) is, unless Ker(ρ[θ]) = \0\ or Ker(ρ[θ]) = \mathbbV, an existentially closed model of TV := T ∪ \\text``V is a vector subspace of \mathbbV''\. In the same vein, we present a criterion for when (M, ρ[θ]) is again an existentially closed model of TC'θ for some C' ∈ C.