# Give an example of a language $B$ is $NL-complete$ where $B^* \in L$

I need to give an example of a language $$B$$ is $$NL-complete$$ where $$B^* \in L$$.

I know $$PATH$$ is $$NL-complete$$ (but not limited to using other languages).

I am clueless about that. I know $$L$$ is not closed under $$^*$$ so there must a language like that.

Any help would be appreciated.

Let $$\Sigma=\{0,1\}$$. Take any NL-complete language $$C \subset \Sigma^*$$ and consider $$B = C \cup \{0,1\}$$. Clearly $$B$$ is still NL-complete (since $$|B \, \Delta \, C| \le 2)$$ and you have $$B^* \supseteq \{0,1\}^* = \Sigma^*$$, i.e., $$B^* = \Sigma^* \in L$$.