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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.

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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., $B^* = \Sigma^* \in L$.

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