{
  "abstract": "Let $θ$ be a regular cardinal, $R$ a ring, and $M$ a left $R$-module. We prove that for every ordinal $α$ there is a set $I_α\\subseteq M^{<θ}$ of size at most $\\beth_α(|R| + θ)$ such that every parameter-free $L_{\\infty,θ}$ formula of rank at most $α$ is equivalent in $M$ to an infinitary Boolean combination of cosets $\\overline{a} + ϕ(M)$, where $\\overline{a} \\in I_α$ and $ϕ$ is an infinitary positive existential formula of rank at most $α$. The main ingredient in the proof is a combinatorial lemma which says that given $κ$ subgroups of an abelian group, there is a set of at most $2^κ$ points which tests whether any family in which each member is either empty or a coset of the corresponding subgroup covers the whole group. The proof proceeds by applying this lemma fiberwise to show that the relevant complete Boolean algebras of positive-existentially definable cosets are closed under projections.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16692",
  "author_total": 1,
  "authors": [
    {
      "name": "Rishi Banerjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dec5t3zge9dke.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16692",
  "fulltext_url": null,
  "id": "dec5t3zge9dke",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16692",
  "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",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7203710385",
  "pdf_url": "https://arxiv.org/pdf/2608.16692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dec5t3zge9dke.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.16692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Infinitary positive existential normal forms for modules",
  "title_text": "Infinitary positive existential normal forms for modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/dec5t3zge9dke",
  "venue": "arXiv",
  "versions": []
}