Ordering of unit systems U and V. A quantity Q ∈ is mapped to and for the representation in the respective unit systems. In the case of U≿V, the preimage is always included in . Only then we can naturally define the mapping .
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 U and U ′. 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 U and V′ ( ) or U ′ and V ( ), which is not shown here.
A hierarchical tree of unit systems. Here, N 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.
Four possible relations between two unit systems U and V: (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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