The algebraic structure of quantity calculus

TL;DR

This paper introduces an algebraic fiber bundle framework to formalize the foundations of quantity calculus, clarifying the relationships among quantities, units, and dimensions within the SI system.

Contribution

It proposes a novel algebraic structure for quantity calculus, providing an axiomatic foundation and classification theorem that formalizes the concepts of quantities and units.

Findings

Provides an algebraic fiber bundle model for quantity calculus

Classifies the algebraic structures underlying quantities and units

Clarifies the relationship between quantities, dimensions, and units

Abstract

An algebraic structure underlying the quantity calculus is proposed consisting in an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product and quotient spaces are considered as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation for quantity calculus and gives complete justification within its framework of the way that quantity calculus is actually performed. It is hoped that this exposition helps to clarify the role of the interviening concepts of quantity, quantity value, quantity dimension and their relation with a system of units, particularly, the SI.