{
  "abstract": "Given a finitely aligned higher-rank $k$-graph $Λ$, Farthing, Muhly and Yeend (Semigroup Forum 2005) constructed an inverse semigroup $S_Λ$ to provide a groupoid model for the $C^*$-algebra of $Λ$. In this paper, we extend some of the existing results related to the semigroup structure of a graph inverse semigroup to the more general setting of the inverse semigroup $S_Λ$. We obtain geometric descriptions of the Green's relations on $S_Λ$, which effectively generalize the existing characterization of Green's relations on a graph inverse semigroup. We provide a simple and useful description of the Cuntz-Krieger ideal of the Booleanization of $S_Λ$ and, based on this, identify a certain $X$-to-join Booleanization of $S_Λ$, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, $S_Λ$ is also ordered by set-inclusion. This helps us to realize $S_Λ$ as a semigroup \\emph{graded-ordered} by $\\mathbb{Z}^k$ and, subsequently, gives rise to a full inverse subsemigroup $S_Λ^h$ of $S_Λ$, which is graded by $\\mathbb{Z}^k$ in the usual sense. We show that the Kumjian-Pask algebra of $Λ$ is graded isomorphic to the quotient of the contracted semigroup algebra of $S_Λ^h$ by the tight ideal. In the final part of the paper, we show that a $k$-graph morphism $p:Λ_1\\longrightarrow Λ_2$ that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from $S_{Λ_1}$ to $S_{Λ_2}$. In view of this, we build a suitable categorical setup to conclude that the isomorphism established between the Kumjian-Pask algebra of $Λ$ and the tight ideal quotient of the semigroup algebra of $S_Λ^h$ is natural.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04398",
  "author_total": 1,
  "authors": [
    {
      "name": "Promit Mukherjee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30mz4856c129q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04398",
  "fulltext_url": null,
  "id": "30mz4856c129q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04398",
  "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.CO"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30mz4856c129q.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Remarks on the Farthing--Muhly--Yeend inverse semigroup of a finitely aligned higher-rank graph",
  "title_text": "Remarks on the Farthing--Muhly--Yeend inverse semigroup of a finitely aligned higher-rank graph",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/30mz4856c129q",
  "venue": "arXiv",
  "versions": []
}