{
  "abstract": "We present an algebraic study of extensions of the implication--negation fragment $\\mathrm{IPC}_{\\to,\\neg}$ of intuitionistic propositional logic. Such extensions by inference rules correspond to subquasivarieties of the variety $\\var{Ho}$ of Hilbert algebras with zero. We describe the lattice of such extensions of $\\mathrm{IPC}_{\\to,\\neg}$ by means of its generators: it is isomorphic to the lattice of upsets of a countable poset, and in particular it is completely distributive. Consequently, $\\var{Ho}$ is not Q-universal, in contrast to several other varieties related to $\\mathrm{IPC}$. Nevertheless, this lattice is very complex: at each suitably defined height it has intervals containing continuum many quasivarieties which are not varieties. To the best of our knowledge $\\mathrm{IPC}_{\\to,\\neg}$ is the only natural fragment of intuitionistic logic known to combine a completely distributive lattice of extensions with continuum many non-axiomatic extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05928",
  "author_total": 2,
  "authors": [
    {
      "name": "Tomasz Kowalski"
    },
    {
      "name": "Katarzyna Słomczyńska"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/exnwc4240gtrx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05928",
  "fulltext_url": null,
  "id": "exnwc4240gtrx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05928",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05928",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/exnwc4240gtrx.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05928",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05928"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Rule extensions of the implication--negation fragment of intuitionistic logic",
  "title_text": "Rule extensions of the implication--negation fragment of intuitionistic logic",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/exnwc4240gtrx",
  "venue": "arXiv",
  "versions": []
}