Many-Sorted Algebraic Semantics for Dimensional Quantity Calculus

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.