{
  "abstract": "In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \\textit{``Strong Reducibilities''} (1981) and \\textit{``Reducibilities''} (1999). The problem asks whether every computably enumerable (c.e.) $tt$-degree contains a c.e.\\ \\textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\\ $tt$-degree that does not contain any c.e.\\ irreducible $m$-degree. Our proof relies on the structural properties of c.e.\\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.03066",
  "author_total": 1,
  "authors": [
    {
      "name": "Patrizio Cintioli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pezqb7qxpkwt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.03066",
  "fulltext_url": null,
  "id": "0pezqb7qxpkwt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.03066",
  "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.LO"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7160425000",
  "pdf_url": "https://arxiv.org/pdf/2605.03066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pezqb7qxpkwt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.03066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.03066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees",
  "title_text": "A Computably Enumerable tt-Degree Without Computably Enumerable Irreducible m-Degrees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0pezqb7qxpkwt",
  "venue": "arXiv",
  "versions": []
}