{
  "abstract": "In this paper, we establish necessary and sufficient representation theorems for the classes $\\widetilde{\\mathcal{U}}_{\\mathrm{top}}$, $\\widetilde{\\mathcal{U}}_{\\mathrm{bot}}$, $\\widetilde{\\mathcal{U}}_{\\mathrm{max}}^{r}$, and $\\widetilde{\\mathcal{U}}_{\\mathrm{min}}^{r}$ of uninorms on bounded trellises, extending the corresponding bounded-lattice results. For a non-extremal neutral element $e$ belonging to no non-trivial cycle, we use a common decomposition of the elements incomparable with $e$; every neutral element of a uninorm is shown to be middle-transitive. Each uninorm in $\\widetilde{\\mathcal{U}}_{\\mathrm{top}}$ (respectively, $\\widetilde{\\mathcal{U}}_{\\mathrm{bot}}$) is represented by an interior operator (respectively, a closure operator), a component uninorm, and an increasing, associative, and commutative operation, with all components uniquely determined. Two further conditions characterize the subclasses $\\widetilde{\\mathcal{U}}_{\\mathrm{top}}^{\\star}$ and $\\widetilde{\\mathcal{U}}_{\\mathrm{bot}}^{\\star}$. For $\\widetilde{\\mathcal{U}}_{\\mathrm{max}}^{r}$ (respectively, $\\widetilde{\\mathcal{U}}_{\\mathrm{min}}^{r}$), we identify the pairs in $I_e^3\\times I_e^3$ whose values lie in $[0,e[\\cup I_e^1$ (respectively, $]e,1]\\cup I_e^2$) or equal $e$. The pairs producing $e$ satisfy a symmetric unique-partner condition, while the remaining values are encoded by a partial operation. Regional order compatibility and explicit associativity conditions then yield necessary and sufficient representations. Finally, we compare the four classes, relate them to the corresponding bounded-psoset classes, prove their bounded-lattice specialization under transitivity, and provide finite examples on proper trellises illustrating the additional compatibility conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08191",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhenyu Xiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33rmmq18pf22m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08191",
  "fulltext_url": null,
  "id": "33rmmq18pf22m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08191",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7220912724",
  "pdf_url": "https://arxiv.org/pdf/2610.08191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33rmmq18pf22m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Representation theorems for four classes of uninorms on bounded trellises",
  "title_text": "Representation theorems for four classes of uninorms on bounded trellises",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/33rmmq18pf22m",
  "venue": "arXiv",
  "versions": []
}