Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular

Question

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular.

These are the things that I would use to start.

• As $L_1L_2$ is regular, then the homomorphism $h(L_1L_2)$ is regular.
• Let $h(L_1) = L_2$ and $h(L_2) = L_1$, then $h(L_1L_2) = L_2L_1$ is regular (we already know that) or $h(L_2) = \epsilon$ and we get $L_1$
• By reflexing, $L_1L_2 = (L_2L_1)^{R}$, same result.

But i don't know how to, for example, intersect something that gives me $L_1$ in order to preserve closure and finally $L_1$ be regular.

Any help?

Answer 1

Here is a counterexample. Let $L_1$ be any language over some alphabet $\Sigma$ containing the empty string, and let $L_2 = \Sigma^*$. Then $L_2 = L_1L_2 = L_2L_1 = \Sigma^*$ are all regular, but $L_1$ need not be, in fact it could even be uncomputable!

Answer 2

Another counterexample (special case of the first answer) is find just one non regular language in specific and another regular language where $L_1 \subseteq L_2$.

Let $L_1 = \{a^{n^2}, n \geq 0\}$ (it includes $\epsilon$) and $L_2 = \{a^n, n \geq 0\}$. It's concatenation $L_1L_2 = L_2L_1 = L_2$ since the subset property established at the beggining. Then $L_1$ is not necessarily regular.