{
  "abstract": "In this paper, we give an affirmative answer to the open question posed by G.Birkhoff and R.S.Pierce in 1956 concerning the existence of a compatible lattice order on the complex field $\\mathbb{C}$. Using valuation rings theory, basis extension in infinite-dimensional vector spaces and extension of field embeddings to algebraic extensions, we obtain that for every real $t>1$ transcendental over $\\mathbb{Q}$, the compatible lattice order can be chosen such that $0\\prec1$, $0\\vee i=t$, and $1\\wedge t=0$. Consequently, $\\mathbb{C}$ admits at least $2^{\\aleph_0}$ distinct compatible lattice orders, each extending the usual order on $\\mathbb{Q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37155",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhengmao He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8zrygkp175adb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37155",
  "fulltext_url": null,
  "id": "8zrygkp175adb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37155",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7224104399",
  "pdf_url": "https://arxiv.org/pdf/2610.37155",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8zrygkp175adb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37155",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37155"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The existence of the compatible lattice order on the complex field $\\mathbb{C}$",
  "title_text": "The existence of the compatible lattice order on the complex field ℂ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/8zrygkp175adb",
  "venue": "arXiv",
  "versions": []
}