{
  "abstract": "The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential \"toggle game\" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős--Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \\ge 10^{5}$ works.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27416",
  "author_total": 1,
  "authors": [
    {
      "name": "Hermann Wilhelm"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cq4y8bxwd7gsd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27416",
  "fulltext_url": null,
  "id": "cq4y8bxwd7gsd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27416",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204521718",
  "pdf_url": "https://arxiv.org/pdf/2608.27416",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cq4y8bxwd7gsd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.27416",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27416"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Refutation of the Non-Cancelling Intersections Conjecture",
  "title_text": "Refutation of the Non-Cancelling Intersections Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/cq4y8bxwd7gsd",
  "venue": "arXiv",
  "versions": []
}