{
  "abstract": "We study polynomial expressions with physical dimensions and the effect of setting an invertible reference quantity equal to one. Dimensions are treated as sorts indexed by a finitely generated free Abelian group D. Using standard categorical constructions, we describe the free D-graded commutative real algebra and its induced finitary monad TD. Monad multiplication is homogeneous polynomial substitution. We identify its Eilenberg–Moore category with the category of D-graded commutative real algebras and give a D-sorted Lawvere theory for the same structure. We then distinguish normalization, coarsening of degree labels, and specialization u=1. For an invertible homogeneous reference u of infinite-order degree g, the specialization map A⟶A/(u−1), with the quotient graded by D/Zg, is injective on each original homogeneous component. Thus, two expressions of the same original degree are equal if and only if their images after specialization are equal, although mixed-degree information may be lost. An equation-of-motion example illustrates dimensional typing, the imposition of a physical relation, and reference specialization. For finitely presented models, the real Nullstellensatz yields existence and joint equation-detection criteria for surjective graded numerical presentations. Models with the same full degree support and real radical relations can differ in both properties. A corollary gives finite algebraic certificates for validity in all such presentations.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Electrical and Electronic Engineering, Shizuoka University, 3-5-1 Johoku, Chuo-ku, Hamamatsu 432-8561, Japan"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Kohei Yaji",
      "orcid": "0000-0001-9900-3049"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fh30bjxsm5kn2.bib",
  "cited_by": 0,
  "doi": "10.3390/math14224128",
  "fulltext_url": null,
  "id": "fh30bjxsm5kn2",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/22/4128",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-10",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/fh30bjxsm5kn2.ris",
  "sources": [
    {
      "first_seen": "2026-11-11T00:21:23Z",
      "native_id": "10.3390/math14224128",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/22/4128"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Many-Sorted Algebraic Semantics for Dimensional Quantity Calculus",
  "title_text": "Many-Sorted Algebraic Semantics for Dimensional Quantity Calculus",
  "type": "peer-reviewed",
  "updated_at": "2026-11-12T00:20:37Z",
  "url": "https://research.draconitic.com/paper/fh30bjxsm5kn2",
  "venue": "Mathematics",
  "versions": []
}