{
  "abstract": "For a row-finite weighted graph $(E,w)$, Preusser showed that the monoid $\\mathcal{V}(L_k(E,w))$ of finitely generated projective modules over the weighted Leavitt path algebra $L_k(E,w)$ is isomorphic to a combinatorially defined weighted graph monoid $\\mathcal{M}(E,w)$. We study two structural properties of $\\mathcal{M}(E,w)$: confluence and cancellativity. We introduce a reduction system on the free commutative monoid presenting $\\mathcal{M}(E,w)$, obtain sufficient conditions for non-confluence by constructing explicit non-confluent triples, and provide a complete confluence characterization for certain classes of weighted graphs. Turning to cancellativity, we work within Preusser's class of weighted graphs satisfying Condition (LPA), for which $L_k(E,w)$ is isomorphic to an unweighted Leavitt path algebra $L_k(F)$ via a two-step construction. We introduce an auxiliary graph associated to the intermediate step of this construction and use it to give a graph-theoretic characterization of when $\\mathcal{M}(E,w)$ is cancellative. Finally, under Condition (LPA), we show that Preusser's construction upgrades to a graded isomorphism $L_k(E,w) \\cong_{\\operatorname{gr}} L_k(F)$ with respect to the standard $\\mathbb{Z}^{λ(E,w)}$-grading of weighted Leavitt path algebras, yielding $\\mathcal{V}^{\\operatorname{gr}}(L_k(E,w)) \\cong \\mathcal{V}^{\\operatorname{gr}}(L_k(F))$ as $\\mathbb{Z}^{λ(E,w)}$-monoids.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31087",
  "author_total": 2,
  "authors": [
    {
      "name": "Rishabh Goswami"
    },
    {
      "name": "Alfilgen Sebandal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ew249jj5w8qbe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31087",
  "fulltext_url": null,
  "id": "ew249jj5w8qbe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31087",
  "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.RA"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7204867231",
  "pdf_url": "https://arxiv.org/pdf/2608.31087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ew249jj5w8qbe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2608.31087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Properties of the $\\mathcal V$-Monoid of Weighted Leavitt Path Algebras",
  "title_text": "Properties of the 𝒱-Monoid of Weighted Leavitt Path Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/ew249jj5w8qbe",
  "venue": "arXiv",
  "versions": []
}