{
  "abstract": "Birkhoff and Pierce asked in 1956 whether the field $\\mathbb{C}$ of complex numbers admits a lattice order compatible with addition and multiplication. We answer the question affirmatively. Using a valuation with value group $\\mathbb{Q}$, we construct a positive cone $P$ by a transfinite recursion of countable ordinal length. At each stage we extend a lattice order on a subspace while preserving its lattice operations in the whole field and maintaining multiplicative closure of the ambient positive cone. These subspaces eventually exhaust $\\mathbb{C}$. The construction depends on choices of a valuation extension, compatible rational powers and vector-space bases. The cone $P$ contains $1$ and satisfies $\\{x\\in P\\setminus\\{0\\}:x^{-1}\\in P\\}=\\mathbb{Q}_{>0}$. The induced order is not total: $i$ is incomparable with $0$. For a chosen transcendental element $t$ and $i^2=-1$, one may prescribe $0\\vee i=(t+i)/2$ and $0\\vee(-i)=(t-i)/2$. Varying the first extension yields exactly $2^{\\mathbf c}$ distinct lattice positive cones on $\\mathbb{C}$, where $\\mathbf c=|\\mathbb{C}|$. The count concerns cones as subsets of $\\mathbb{C}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38300",
  "author_total": 2,
  "authors": [
    {
      "name": "Hanzhen Bao"
    },
    {
      "name": "Yuehui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78k42mbr8a9bw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38300",
  "fulltext_url": null,
  "id": "78k42mbr8a9bw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38300",
  "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.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38300",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78k42mbr8a9bw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38300",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38300"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A lattice order on the complex field",
  "title_text": "A lattice order on the complex field",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/78k42mbr8a9bw",
  "venue": "arXiv",
  "versions": []
}