Homomorphism. Equation.

Question

Let $h$ be a homomorphism monoid $M =\{0,1\}^*$ $h: M \to M, h(0) = 1, h(1) = 010$ And it is true: $h^3(1^+) = (h(1)h(1)h(1) )^+ $ I don't understand this equation.

Answer 1

After the clarifications in the comments below the question, I understand your question as follows: is it true that $\Big( h(1^+) \Big)^3 = \Big( h(1)^3 \Big) ^+$? For mathematicians not accustomed with notations from formal languages, this can be reread as: is it true that for every $n \ge 1$ there exist $m \ge 1$ such that $\Big( h(1^n) \Big)^3 = \Big( h(1)^3 \Big) ^m$? Let us prove the statement by induction on $n$: clearly it is true for $n=1$ (just choose $m=1$); for $n>1$

$$\Big( h(1^{n+1}) \Big)^3 = \Big( h(1^n 1) \Big)^3 = \Big( h(1^n) h(1) \Big)^3 = h(1^n) h(1) h(1^n) h(1) h(1^n) h(1)= h(1)^m h(1) h(1)^m h(1) h(1)^m h(1) = h(1)^{3m+3} .$$

In fact, one can do even better: the above proof shows that the dependence of $m$ on $n$ can be given recursively by $m(n+1) = 3 m(n), \space m(1)=1$. This gives $m(n) = 3^{n-1}$, so we can be even more precise: $\Big( h(1^n) \Big) ^3 = \Big( h(1) \Big)^{3^{n-1}}$.