Closure set of closure set intersection - formal languages Can we say...
Given L1 and L2.
Is the following true?
$$
L_{1}^{*} \cap L_{2}^{*} = (L_{1}^{*} \cap L_{2}^{*})^{*}
$$
I think it is true but I can't be sure. What permutation of either wouldn't be contained in each other. Sorry I don't know how to format on Math Exchange to make things look better.
-Steve

Let $L$ be a language of the free monoid $A^*$. The answer easily follows form the fact that $L = L^*$ if and only if $L$ is a submonoid of $A^*$.
Now $L_1^*$ and $L_2^*$ are submonoids of $A^*$ and hence their intersection is a submonoid of $A^*$. Thus $L_1^* \cap L_2^*$ is a submonoid of $A^*$ and it is equal to its own star.
