{
  "abstract": "We define a binary operation $T_1\\star T_2$ on the fuzzy unit interval $\\I$ by extending the natural action of $T_1$ and $T_2$ on step functions. When both $T_1$ and $T_2$ are left-continuous, this operation coincides with the tensor product of the corresponding quantales and therefore yields a t-norm on $\\I$. Without left-continuity, the situation becomes more subtle. We prove that if at least one of the two t-norms is continuous, then $T_1\\star T_2$ is still a t-norm on $\\I$. We also show by counterexample that continuity cannot simply be dropped. The operation $\\DN$ fails to be associative. On the other hand, $T_D\\star T_D$ is a t-norm despite $T_D$ being neither continuous nor left-continuous, indicating that continuity is not a necessary condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40678",
  "author_total": 3,
  "authors": [
    {
      "name": "Rongjun Wu"
    },
    {
      "name": "Jie Sun"
    },
    {
      "name": "Hongliang Lai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6tsw8hea5zhm7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40678",
  "fulltext_url": null,
  "id": "6tsw8hea5zhm7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40678",
  "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.GM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40678",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6tsw8hea5zhm7.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40678",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40678"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms",
  "title_text": "Beyond left-continuity: t-norms on the fuzzy unit interval induced by pairs of t-norms",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/6tsw8hea5zhm7",
  "venue": "arXiv",
  "versions": []
}