Many-Sorted Algebraic Semantics for Dimensional Quantity Calculus
Year: 2026
Type: article
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.
Source: Mathematics
DOI: 10.3390/math14224128
Author: Kohei Yaji
Institution: Shizuoka University
Language: English
Subfield: Software
Field: Computer Science
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
OpenAlex ID: W7225745027
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