{
  "abstract": "We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid $H$ satisfying $1<ρ(H)<ρ(G)$, where $ρ$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter $μ(G)$ and prove that $ρ(XY)\\geq \\min\\{μ(G),ρ(X)+ρ(Y)-1\\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $ρ(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18340",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohsen Aliabadi"
    },
    {
      "name": "Jozsef Losonczy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drskwb7e5en2k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18340",
  "fulltext_url": null,
  "id": "drskwb7e5en2k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18340",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7203915390",
  "pdf_url": "https://arxiv.org/pdf/2608.18340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drskwb7e5en2k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.18340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Matchings and product growth in modular abelian independence groups",
  "title_text": "Matchings and product growth in modular abelian independence groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/drskwb7e5en2k",
  "venue": "arXiv",
  "versions": []
}