Physical quantities as a partially additive field

TL;DR

The paper develops an axiomatic framework for physical quantities by introducing a partially additive field $F$ in which addition is partial. It shows that the dimensionless subfield $F_1$ is a field and that every quantity can be uniquely decomposed as a dimensionless value times a unit from a unit system $U$, enabling standard arithmetic when the unit system is a group. It further provides two sufficient conditions for the existence of a coherent unit system (no dimensionful roots of dimensionless quantities and cotorsion of the dimensionless subgroup) and extends the construction to fieldoids, a union of disjoint PAFs. Overall, the work unifies quantities, values, units, and dimensions within a single algebraic structure and highlights how partial operations can elegantly model physical reality, with potential generalizations to ordered or non scalar quantities.

Abstract

We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually summable elements (similar to physical dimensions), dimensionless elements (those summable with 1) form a field, and every element can be uniquely represented as a product of a dimensionless element and any non-zero element of the same dimension (a unit). We also discuss the conditions for the existence of a coherent unit system. In contrast to previous works, our axiomatization encompasses quantities, values, units, and dimensions in a single algebraic structure, illustrating that partial operations may provide a more elegant description of the physical world.

Theorems & Definitions (50)

• Definition 1
• Definition 2
• Lemma 3
• proof
• Corollary 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7