{
  "abstract": "We study the functorial and categorical structure of the canonical rigid direct-system representation of finite local-unit-aligned totally ordered monoids. The local-unit map $τ$ induces a canonical $τ$-multiplication-coherent decomposition into component monoids, and the associated representation reconstructs the original ordered monoid from a finite chain-indexed rigid direct system whose proper transition maps are unit-constant. The present paper identifies the morphism classes for which this representation is categorical. First, we prove an equivalence between finite local-unit-aligned totally ordered monoids with strict block morphisms and rigid direct systems with directed-order-compatible system morphisms. Second, we prove an intrinsic equivalence for $τ$-compatible homomorphisms, that is, isotone unital homomorphisms commuting with the local-unit map. In this second setting, distinct positive idempotents and hence distinct canonical components may collapse to a single target component; on the direct-system side this is represented by non-injective isotone index maps together with component maps satisfying the corresponding collapse and absorption axioms. Thus the canonical rigid direct-system representation is functorial both for strict block morphisms and for intrinsic $τ$-compatible homomorphisms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20233",
  "author_total": 1,
  "authors": [
    {
      "name": "Sándor Jenei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ncktc2h6vv0d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20233",
  "fulltext_url": null,
  "id": "0ncktc2h6vv0d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20233",
  "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.CT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "1703|semigroups and automata theory"
    ]
  ],
  "openalex": "W7213583508",
  "pdf_url": "https://arxiv.org/pdf/2609.20233",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ncktc2h6vv0d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20233",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20233"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids",
  "title_text": "Categorical forms of the rigid direct-system representation for finite local-unit-aligned totally ordered monoids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/0ncktc2h6vv0d",
  "venue": "arXiv",
  "versions": []
}