{
  "abstract": "This paper investigates the completeness properties of the fuzzy order on the fuzzy number space $\\mathbb{E}^1$ within the framework of $[0,1]$-enriched categories. \\(S\\)-convergence under the three fundamental continuous t-norms is characterized, and inclusion relations among the induced open ball topologies are established. It is shown that $(\\mathbb{E}^1,P)$ is Cauchy complete if and only if the t-norm is not isomorphic to the Łukasiewicz t-norm, whereas $(\\mathbb{E}^1,P)$ fails to be Smyth complete for every continuous t-norm. Consequently, the symmetrization $(\\mathbb{E}^1,S)$ is Smyth complete if and only if the underlying continuous t-norm is not isomorphic to the Łukasiewicz t-norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16052",
  "author_total": 1,
  "authors": [
    {
      "name": "Mingchun Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1d7mg2zq76rq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16052",
  "fulltext_url": null,
  "id": "a1d7mg2zq76rq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16052",
  "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",
      "math.GM"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7213405943",
  "pdf_url": "https://arxiv.org/pdf/2609.16052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1d7mg2zq76rq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Completeness of the Fuzzy Order on Fuzzy Numbers",
  "title_text": "Completeness of the Fuzzy Order on Fuzzy Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/a1d7mg2zq76rq",
  "venue": "arXiv",
  "versions": []
}