{
  "abstract": "We give an explicit countably infinite orthomodular lattice generated by three specified elements and a direct argument that it admits no injective homomorphism into any $σ$-complete orthomodular lattice. In particular, it does not embed into any set-sized complete orthomodular lattice. The source is a generated subalgebra of a cofinite reduced product of finite-dimensional real subspace lattices. A nonzero finite term $a$ is orthogonal to the orbit of a terminal line under every finite word in three Sasaki projections, while another finite term $X$ satisfies $π_aπ_X(t)=a$. If the image of this countable orbit has a supremum in a target orthomodular lattice, Sasaki adjunction forces that supremum to commute with the generator images. This forces the image of $a$ to vanish. The proof imposes no atomicity, modularity, Hilbert representation, cardinality bound on the target, or preservation of source infinitary operations. All target lemmas are derived from the orthomodular axioms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14859",
  "author_total": 1,
  "authors": [
    {
      "name": "Joaquim Reizi Higuchi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2h0bse6a5gd0q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14859",
  "fulltext_url": null,
  "id": "2h0bse6a5gd0q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14859",
  "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.LO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7221755144",
  "pdf_url": "https://arxiv.org/pdf/2610.14859",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2h0bse6a5gd0q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14859",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14859"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Countable Three-Generated Orthomodular Lattice with No $σ$-Complete Embedding",
  "title_text": "A Countable Three-Generated Orthomodular Lattice with No σ-Complete Embedding",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/2h0bse6a5gd0q",
  "venue": "arXiv",
  "versions": []
}