Let $A$ be an alphabet. Let $X,L \subset A^*$ $L$ is regular. Let $$X^{-1}L := \{ w \in A^* \mid \exists x \in X\ \ xw \in L \} $$ $$LX^{-1} := \{ w \in A^* \mid \exists x \in X\ \ wx \in L \} $$
As we know, languages are free monoids. Is there any isomorphism from $X^{-1} L \to L X^{-1} $?
I was trying to find something but I got stuck. Thanks in advance.
Hint: look at $L=a^*b$ and $X=\{b\}$ and calculate $LX^{-1}$ as well as $X^{-1}L$.