Neutral element for a Cantor Set.

Question

On Wiki I found the following statement:

$T_L$ and $T_R$ together with function composition forms a monoid.

I am able to prove the associativity of the composition operation, but what will be a neutral element for the operation?

Thank you.

Answer 1

As Mees de Vries says, it should really say "generates" rather than "forms," and you're exactly right about why: there's no identity (= neutral) element. However, if we broaden our horizons a bit, it can be true as written! Similarly to how the empty sum, empty product,and various other "empty operations" can be defined, there is a natural notion of "composition of no functions" - namely, the empty composition is the identity function, and this accounts for the "missing" identity element in our monoid here.

Let's think about a single function first. Composition satisfies the additivity rule $f^m\circ f^n=f^{m+n}$ for $m,n$ natural numbers $>0$, where "$f^k$" denotes the function gotten by composing $f$ with itself $k$ times. Based on this there is a unique way to define $f^0$ so as to follow the pattern: $f^0$ should be the identity map.

Now you're looking at a slightly more complicated situation: you have two functions involved, and can compose them with each other. Here natural numbers $>0$ get replaced by finite binary strings of length $>0$: e.g. $$T_LT_RT_L$$ is represented by the string $010$ (arbitrarily setting $L=0,R=1$). In place of additivity, we have a concatenation rule: the composition of the functions represented by the strings $\sigma$ and $\tau$ is the function represented by the string $\sigma\tau$. Now, what should the function corresponding to the empty string be, based on this rule?