2025/12/20 by Leon Chini, Chini, Leon
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Algebra over a field #Algebraic number #Endomorphism #Homotopy and Cohomology in Algebraic Topology #Injective function #Set (abstract data type) #Space (punctuation) #Vector space #math.LO
paper · pdf · doi:10.48550/arxiv.2512.18327
openalex publication_date 2025/12/20 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
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 \mathbbV in every model, we set Tθ:= T ∪ \\text``θ is a K-endomorphism of \mathbbV''\. We previously defined a family \TCθ: C ∈ C\ of extensions of Tθ 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, and all the ρ[θ]'s and η[θ]'s are polynomials over K with θ plugged in. Notice that properties such as θ2 - 2Id = 0 or ``ρ[θ] is injective for every ρ∈ K[X] ∖ \0\'' can be expressed in such a manner. We also presented a sufficient condition that implies that every TCθ has a model companion TθC. Under this condition, we characterize all definable sets in TθC and study the completions of TθC as well as the algebraic closure. If T is o-minimal and extends Th(ℝ, <), we prove that TθC has an o-minimal open core.