{
  "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 $\\{T^C_θ: C \\in \\mathcal{C}\\}$ of extensions of the theory $T_θ:= T \\cup \\{\\text{``$θ$ is an endomorphism of $\\mathbb{V}$''}\\}$ that parameterizes all consistent extensions of the form $$ T_θ\\cup \\left\\{\\sum\\nolimits_{k}\\bigcap\\nolimits_{l}\\operatorname{Ker}(ρ_{j, k, l}[θ]) = \\sum\\nolimits_{k}\\bigcap\\nolimits_{l} \\operatorname{Ker}(η_{j, k, l}[θ]) : j \\in \\mathcal{J}\\right\\}, $$ 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 $\\mathcal{J}$ 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 $\\operatorname{TP}_2$ and $\\operatorname{SOP}$, and are $\\operatorname{NATP}$, as well as distal non-o-minimal theories of various $\\operatorname{dp}$-ranks with and without the exchange property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20184",
  "author_total": 1,
  "authors": [
    {
      "name": "Leon Chini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/65pzfkk96cyw4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20184",
  "fulltext_url": null,
  "id": "65pzfkk96cyw4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20184",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.20184",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/65pzfkk96cyw4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20184",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20184"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Model Theory of Generic Vector Space Endomorphisms V: The o-Minimal Case",
  "title_text": "Model Theory of Generic Vector Space Endomorphisms V: The o-Minimal Case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/65pzfkk96cyw4",
  "venue": "arXiv",
  "versions": []
}