Mathematical structure of unit systems

TL;DR

This paper explores the mathematical structure of unit systems, revealing a hierarchical organization based on preorder and equivalence classes, which aids systematic comparison and classification of different unit systems.

Contribution

It introduces a formal mathematical framework using preorder and equivalence classes to analyze and organize unit systems systematically.

Findings

Unit systems form a preorder structure allowing transfer relations.

Equivalence classes (EUS) unify representations of physical quantities.

Unit systems are organized into a hierarchical tree based on their relations.

Abstract

We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit systems, there exists a relation of preorder such that one unit system is transferable to the other unit system. The transfer (or conversion) is possible only when all of the quantities distinguishable in the latter system are always distinguishable in the former system. By utilizing this structure, we can systematically compare the representations in different unit systems. Especially, the equivalence class of unit systems (EUS) plays an important role because the representations of physical quantities and equations are of the same form in unit systems belonging to an EUS. The dimension of quantities is uniquely defined in each EUS. The EUS's form a partially ordered set. Using these mathematical structures, unit systems and EUS's are systematically classified and organized as a hierarchical tree.