{
  "abstract": "We establish necessary and sufficient conditions under which a binary operation on a bounded trellis is a proper nullnorm. The representation combines a t-conorm on the lower interval, a t-norm on the upper interval, two order-preserving maps, and a commutative, increasing function on $I_a^3\\times I_a^3$, where $I_a^3$ consists of the elements incomparable with the absorbing element $a$ that neither reach $a$ nor are reachable from $a$. Unlike earlier range-restricted constructions, this function may take values anywhere in the trellis. To preserve associativity for such unrestricted values, we introduce a mixed interaction function that evaluates every pair with at least one component in $I_a^3$. We also derive the specializations in which this region is empty or consists of a single element, including an exact description of the admissible value in the singleton case. A five-element lattice example shows that the mixed associativity condition is independent of the remaining hypotheses, while a fourteen-element nontransitive trellis example demonstrates the necessity of allowing the unrestricted range. Finally, the principal range-restricted subclasses and the bounded-lattice case are recovered as specializations of the general representation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14509",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhenyu Xiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8cspqst1m4rjn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14509",
  "fulltext_url": null,
  "id": "8cspqst1m4rjn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14509",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7213222991",
  "pdf_url": "https://arxiv.org/pdf/2609.14509",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8cspqst1m4rjn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14509",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14509"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A complete representation theorem for nullnorms on bounded trellises",
  "title_text": "A complete representation theorem for nullnorms on bounded trellises",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8cspqst1m4rjn",
  "venue": "arXiv",
  "versions": []
}