# Is there a context free, non-regular language $L$, for which $L^*$ is regular?

I know that there are non-regular languages, so that $L^*$ is regular, but all examples I can find are context-sensitive but not context free.

In case there are none how do you prove it?

• Can be answered with the same techniques as cs.stackexchange.com/questions/1549 – sdcvvc May 25 '12 at 20:54
• Hint: all languages which contain the alphabet have a very simple Kleene closure. – Raphael Dec 15 '14 at 22:58

$L = \{a^n b^n \mid n\in\mathbb{N}\}$ is context-free but not regular (classical example). So is $L' = \{a^n b^n \mid n\in\mathbb{N}\} \cup \{a,b\}$.
$L'^\ast = \{a,b\}^\ast$ is regular.
• $L' = L$, actually... – vonbrand Nov 30 '16 at 1:48