Physical quantities as a partially additive field

TL;DR

This paper introduces a new algebraic structure called a partially additive field, which models physical quantities by allowing addition to be a partial operation, unifying quantities, units, and dimensions.

Contribution

It generalizes the concept of a field to include partial addition, providing a unified algebraic framework for physical quantities, units, and dimensions.

Findings

Elements form subsets of mutually summable elements

Dimensionless elements form a field

Unique representation of quantities as product of a unit and a dimensionless element

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.