# If L * is regular, then is L regular?

I've been trying to find an answer and I'm getting conflicting answers so I'm not sure. I know the converse is true, that if L is regular then L * is regular when closed.

I believe that if L * is regular, then L is regular, because the subset of L * must be regular and L is part of that subset.

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If L * is regular, then L is not necessarily regular.

Here's one possible example. Let & Sigma; = {a} and consider the language L = {a ^{2 n} | n & in; N}. This language is not regular, and you can prove it by using either the Swapping Lemma for regular languages or the Mihill-Nerod Theorem. However, L * is a *, which is regular. To see this, note that since L contains a string a, L * contains all strings of the form a ^{n} for any natural n.

Hope this helps!

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