L $ \subseteq \{a\}^* $ => $L^*$ is regular I having trouble showing that if $L \notin REG $ => $ (L^*)= \hat L \in REG $. I know that if $ | \Sigma/ \sim_\hat L | < \infty => L \in REG $ so there must be a way to tell if w $ \in \\{a \\}^* $ is in one of the classe $[a]_{\sim_\hat L} $ = {w $ \in \\{ a\\}^* | a \sim_\hat L w $}. Can you help me proofing this Lemma ?

**Hint:**

* Because $L \subseteq \\{\mathtt{a}\\}^*$, the only thing that matters are the lengths of words.
* In fact, we only care about lengths after we cut off an arbitrairly long initial segment (because any finite subset of $\\{\mathtt{a}\\}^*$ is regular).
* WLOG we can assume $\varepsilon\
otin L$ (the empty word does not change anything for $L^*$).
* Let $\mathtt{a}^n$ be the shortest word in $L$. Now consider lengths of words of $L^*$ modulo $n$. We set $w_0 = \mathtt{a}^n$, and define $w_i$ as the shortest word of $L^*$ which length is $i$ modulo $n$ or the empty word if no such word belongs to $L^*$, i.e. \begin{align*}A_i &= \\{k\in\mathbb{N}\mid \mathtt{a}^k\in L^*, k\bmod n= i\\}\\\w_i &= \begin{cases}\mathtt{a}^{\min A_i}&\text{ if } A_i\
eq\varnothing\\\\\varepsilon&\text{ otherwise}\end{cases}\end{align*}
* Prove that for some finite language $I$ we have $L^* \subseteq I\cup\\{w_0,w_1,\ldots,w_{n-1}\\}^*$.



I hope this helps $\ddot\smile$