4.4.  Relationships between reﬁnement properties

Deﬁnition 16 contains many reﬁnement properties so we will attempt to relate these properties to each other in this 
section. A partial summary of the results can be found in Fig. 4. In the later parts of the section, we will also consider some 
relations between reﬁnements properties and the transformation properties from Deﬁnition 9. In particular, we show that 
the hierarchies in Fig. 4 collapse when combined with certain transformation properties. The following examples are used 
in the forthcoming proofs and are collected here for reference.

f (4) = 3, R(a, a) and τ = (cid:3) f , R(cid:4).
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