{
  "abstract": "We develop a logarithmic framework to study the lattice of subgroups of the additive group of rational numbers. By encoding positive rationals via their prime exponent sequences, one obtains a bijection between $\\mathbb{Q}^+$ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of $(\\mathbb{Q},+)$ through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in $\\mathbb{Z}\\cup \\{- \\infty\\}$. Within this framework, subgroup inclusion, sum, product, and intersection admit simple coordinatewise descriptions, providing a transparent interpretation of the subgroup lattice and recovering several classical results such as the subgroup classification up to isomorphism. Exploiting this perspective, we obtain a complete characterization of the intersection configurations realizable in subgroups of $(\\mathbb{Q},+)$. For rank configurations, realizability is characterized by the decreasing condition together with restrictions on the possible ranks and on the minimal zero sets, and, in the finite-support case, by an additional cardinality condition. For binary configurations, this reduces to saying that every decreasing configuration is realizable in the infinite-support case, whereas in the finite-support case the mentioned cardinality condition is required.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37357",
  "author_total": 3,
  "authors": [
    {
      "name": "Jordi Delgado"
    },
    {
      "name": "Antoni Massegú"
    },
    {
      "name": "Enric Ventura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rrz9gn53tj0j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37357",
  "fulltext_url": null,
  "id": "0rrz9gn53tj0j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37357",
  "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.GR"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214989199",
  "pdf_url": "https://arxiv.org/pdf/2609.37357",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rrz9gn53tj0j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37357",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37357"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Additive subgroups of Q: logarithmic description and intersection configurations",
  "title_text": "Additive subgroups of Q: logarithmic description and intersection configurations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/0rrz9gn53tj0j",
  "venue": "arXiv",
  "versions": []
}