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Mathematical structure of unit systems

### 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.

© 2013 AIP Publishing LLC

Received 21 October 2012
Accepted 08 April 2013
Published online 03 May 2013

Acknowledgments:
The authors would like to thank S. Tanimura, Y. Nakata, and S. Tamate for their helpful discussions. The present study was supported in part by Grants-in-Aid for Scientific Research (22109004 and 22560041) and by the Global COE program “Photonics and Electronics Science and Engineering” of Kyoto University.

Article outline:

I. INTRODUCTION
II. BASICS OF UNIT SYSTEMS
A. Ensembles of quantities
B. Representation of quantities with base units
III. PREORDER OF UNIT SYSTEMS
A. Unit-system dependent distinguishability of quantities
B. Transferability of unit systems
IV. CONVERSION OF UNIT SYSTEMS
A. Mapping from one unit system to another
B. Transfer matrix
C. Composition of transformations
V. QUANTITIES IN EQUIVALENT UNIT SYSTEMS
A. Equivalence class of unit systems
B. Relation among equivalence classes of unit systems — partial order
C. Dimension of e-quantities
D. Dimensional analysis and the Buckingham Pi-theorem
VI. STANDARD FORM OF TRANSFORMATION
A. Decomposition of the transfer matrix
B. Quantities transferred to unity
VII. COMPARISON OF UNIT SYSTEMS WITH NORMALIZED QUANTITIES
A. Normalized quantities
B. Comparison of incomparable unit systems
VIII. PRACTICAL UNIT SYSTEMS
A. Examples of the use of normalized quantities
IX. RELATIONS BETWEEN REAL UNIT SYSTEMS
A. Example 1
B. Example 2
C. MKSA to CGS emu
D. MKSA to CGS esu
E. MKSA to a symmetric three-base unit
F. The modified Heaviside-Lorentz system
G. Examples of dimensions
X. CONCLUSION

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2013-05-03

2016-10-23

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