{
  "abstract": "Given a noncomputable, computably enumerable set $D$, we construct a computable structure $G$ such that any two computable copies of $G$ are computably isomorphic while there are two $D$-computable copies of $G$ that are not $D$-computably isomorphic. In other words, $G$ is computably categorical, but not computably categorical relative to $D$. Conversely, we construct a computable structure that is not computably categorical, but is computably categorical relative to $D$. These results differ from those in the literature regarding computable categoricity and its relativiziations because the degree $D$ is given instead of constructed. Our work negatively answers a question of Downey, Harrison-Trainor and Melnikov in the computably enumerable case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21047",
  "author_total": 3,
  "authors": [
    {
      "name": "David Gonzalez"
    },
    {
      "name": "Java Darleen Villano"
    },
    {
      "name": "José Jeremías Valenzuela Morales"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/872fn195yga76.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21047",
  "fulltext_url": null,
  "id": "872fn195yga76",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21047",
  "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.LO"
    ],
    [
      "openalex",
      "1703|Computability, Logic, AI Algorithms"
    ]
  ],
  "openalex": "W7213950966",
  "pdf_url": "https://arxiv.org/pdf/2609.21047",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/872fn195yga76.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21047",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21047"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Switching Categoricity Behavior with a Given Degree",
  "title_text": "Switching Categoricity Behavior with a Given Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/872fn195yga76",
  "venue": "arXiv",
  "versions": []
}