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and
:<math>c\left(\frac{x+2}{3}\right) = \frac{1+c(x)}{2}</math>
The magnifications can be cascaded; they generate the [[dyadic monoid]]. This is exhibited by defining several helper functions. Define
:<math>r(x)=1-x</math>
The first self-symmetry can be expressed as
:<math>r\circ c = c\circ r</math>
where the symbol <math>\circ</math> denotes function composition. For the left and right magnifications, write the left-mappings
:<math>L_D(x)= \frac{x}{2}</math>
and
:<math>L_C(x)= \frac{x}{3}</math>
Then the Cantor function obeys
:<math>L_D \circ c = c \circ L_C</math>
Similarly, define the right mappings as
:<math>R_D(x)= \frac{1+x}{2}</math>
and
:<math>R_C(x)= \frac{2+x}{2}</math>
Then, likewise,
:<math>R_D \circ c = c \circ R_C</math>
== Generalizations ==
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