Model Theory of Generic Vector Space Endomorphisms V: The o-Minimal Case
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 ∅-defines an infinite K-vector space 𝕍. In previous work, we introduced a family {TCθ: C ∈ 𝒞} of extensions of the theory Tθ:= T ∪ {``θ$ is an endomorphism of 𝕍''}\}$ that parameterizes all consistent extensions of the form Tθ∪ {∑k⋂lKer(ρj, k, l[θ]) = ∑k⋂l 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 study the case where T is an o-minimal expansion of the theory of ordered groups. Doing so, we obtain a new family of theories that have TP2 and SOP, and are NATP, as well as distal non-o-minimal theories of various dp-ranks with and without the exchange property.
Related
- Topological Vector Group Topologies Between the Minimal Topology and the Usual Topology on the Real Line
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- Root vector localization and minimal nilpotent finite W-algebras
BibTeX
@misc{chini2026model,
title = {{Model Theory of Generic Vector Space Endomorphisms V: The o-Minimal Case}},
author = {Leon Chini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20184},
eprint = {2609.20184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20184},
}
RIS
TY - UNPB
TI - Model Theory of Generic Vector Space Endomorphisms V: The o-Minimal Case
AU - Leon Chini
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20184
UR - https://arxiv.org/abs/2609.20184
AB - 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 ∅-defines an infinite K-vector space 𝕍. In previous work, we introduced a family {T^C_θ: C ∈ 𝒞} of extensions of the theory T_θ:= T ∪ {``θ$ is an endomorphism of 𝕍''}\}$ that parameterizes all consistent extensions of the form T_θ∪ {∑_k⋂_lKer(ρ_j, k, l[θ]) = ∑_k⋂_l Ker(η_j, 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 T^C_θ has a model companion Tθ^C. In this paper, we study the case where T is an o-minimal expansion of the theory of ordered groups. Doing so, we obtain a new family of theories that have TP_2 and SOP, and are NATP, as well as distal non-o-minimal theories of various dp-ranks with and without the exchange property.
ER -