# Inverse Homomorphisms and Kleene star

The exercise is to prove or give a counterexample to the following proposition with $L \subseteq \Gamma^*$ regular and $h: \Gamma \to \Sigma^*$ a homomorphism.

Is there any regular language $L'$ such that $h^{-1}(L^*) = L'^*$ ?

I know that it isn't that easy to see, since $h^{-1}(L^*) \not= h^{-1}(L)^*$ in general with $L = \{a\}$ and $h : \{a,b\} \to \{a\}^*$ with $h(a) = a, h(b) = aa$. It is $h^{-1}(L^*) = \{a,b\}^* \not= \{a\}^* = h^{-1}(L)^*$.

I still fail to deliver a proper argument to prove it.

Since you solved the first question, let me answer the second one. If you don't mind, I will use $h$ instead of $h'$ for simplicity. Let $L$ be a regular language and let $K = h^{-1}(L^*)$. Since regular languages are closed under inverses of homorphisms, $K$ is regular. Moreover, if $u, v \in h^{-1}(L^*)$, then $h(u), h(v) \in L^*$ and hence $h(uv) = h(u)h(v) \in L^*$. It follows that $uv \in h^{-1}(L^*)$ and thus $K = K^*$. Consequently, $h^{-1}(L^*) = K^*$, which answers your question.