{
  "abstract": "The complete-lattice tensor product $\\mathbb{E}=[0,1]\\otimes[0,1]$ arises naturally in the algebra of truth values for type-2 fuzzy sets. For a monotone surjective map $S$ that is continuous on $[0,1[^2$ and a monotone map $T$ having $1$ as identity, we consider the generalized supremum convolution \\[ (f\\otimes_{_{S,T}}g)(x)=\\sup_{S(u,v)=x}T(f(u),g(v)) \\] on $\\mathbb{E}$. Rather than assuming classical algebraic structures for $S$ and $T$ from the outset, we ask when they are forced by requiring the induced convolution itself to be a t-conorm. We prove that $\\otimes_{_{S,T}}$ is a t-conorm on $\\mathbb{E}$ if and only if $S$ is a continuous t-conorm and either $S=\\max$ or $T$ is a right-continuous t-norm; in the exceptional case $S=\\max$, the convolution is simply the pointwise maximum for every admissible $T$. The same convolution mechanism also yields an order-dual description of triangle functions on the space of distance distribution functions $Δ^+$. Using an order-reversing isomorphism between $Δ^+$ and a boundary variant of $\\mathbb{E}$, we show that $τ_{_{T^*,L}}$ is a triangle function precisely when $L$ is a continuous t-conorm with no zero divisor and, unless $L=\\max$, $T^*$ is a left-continuous t-conorm; moreover, when $L\\neq\\max$, continuity of $τ_{_{T^*,L}}$ is equivalent to continuity of $T^*$. This resolves the remaining $τ_{_{T^*,L}}$ part of the Schweizer--Sklar problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40659",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengyu Luo"
    },
    {
      "name": "Lili Shen"
    },
    {
      "name": "Jie Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5zx6t75mz9z8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40659",
  "fulltext_url": null,
  "id": "5zx6t75mz9z8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40659",
  "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.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5zx6t75mz9z8f.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions",
  "title_text": "Generalized supremum convolutions on the tensor product of unit intervals: t-conorms and triangle functions",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/5zx6t75mz9z8f",
  "venue": "arXiv",
  "versions": []
}