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Arbitrary finite-length strings in the letters L and R correspond to the [[dyadic rationals]], in that every dyadic rational can be written as both <math>y=n/2^m</math> for integer ''n'' and ''m'' and as finite length of bits <math>y=0.b_1b_2b_3\cdots b_m</math> with <math>b_k\in \{0,1\}.</math> Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.
Some notational rearrangements can make the above slightly easier to express. Let <math>g_0</math> and <math>g_1</math> stand for L and R. Function composition extends this to a [[monoid]], in that one can write <math>g_{010}=g_0g_1g_0</math> and generally, <math>g_Ag_B=g_{AB}</math> for some binary strings of digits ''A'', ''B'', where ''AB'' is just the ordinary [[concatenation]] of such strings. The dyadic monoid ''M'' is then the monoid of all such finite-length left-right moves. Writing <math>\gamma\in M</math> as a general element of the monoid, there is a corresponding self-symmetry of the Cantor function:
:<math>\gamma_D\circ c= c\circ \gamma_C</math>
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