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Mathematical structure of unit systems
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Image of FIG. 1.
FIG. 1.

Ordering of unit systems and . A quantity is mapped to and for the representation in the respective unit systems. In the case of , the preimage is always included in . Only then we can naturally define the mapping .

Image of FIG. 2.
FIG. 2.

Two equivalence classes of unit systems (EUS's) satisfying , with the mapping . Each EUS contains equivalent unit systems as , . There are invertible mappings between any pair of unit systems in an EUS, for example, and between and . There is a (one-way) mapping from any unit system in to any unit system in . These mappings ( and in this example) are related as . Note that there are also mappings between and ′ ( ) or and ( ), which is not shown here.

Image of FIG. 3.
FIG. 3.

A hierarchical tree of unit systems. Here, is the number of base units. Arrows indicate transferability “≻,” and the associated quantity is considered to be unity on the transfer. Dashed boxes represent EUS's, and the four- and two-base unit systems listed are equivalent within each group, whereas the three-base unit systems are all incomparable.


Generic image for table
Table I.

Four possible relations between two unit systems and : (strictly) transferable to (≻), (strictly) transferable from (≺), equivalent (∼), and incomparable (∥). The relations between and , which are the numbers of base units, are also listed.


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Scitation: Mathematical structure of unit systems