# Can a non-regular language $L$ have a non regular $L^*$?

I have been looking around and i cant seem to find an example of such case that a non-regular $$L$$ has a non regular $$L^*$$. Is it possible? If so, can you provide an example of such case please?

• Do you know any non regular language $L$? Have you tried with one of them? – Nathaniel May 18 at 22:08
• I do, for example, L = {a^nb^n | n e N}. I have tried but am still having troubles with this topic. – Justin Park May 18 at 22:20
• I am sure that you can succeed in proving that for this language, $L^*$ is not regular, since the same usual proof (with pumping lemma) works. – Nathaniel May 18 at 22:21
• Thank you guys so much! – Justin Park May 18 at 22:23
• For your information: if $L$ is a language over a single letter alphabet, then $L^*$ is always regular, whetever the complexity of $L$. But that is a special case of course. – Hendrik Jan May 20 at 1:39

Take the Dyck language $$D$$. It is context-free but not regular and satisfies $$D^* = D$$.