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Given a string, take all of its substrings (including the empty string). For example, given $abc$, we can form a set $\{\emptyset, a, b, c, ab, bc, abc\}$.

Given a regular language, take all the substrings of each of its strings, and form a language thereof. I wonder if the formed language is still regular?

Btw, is there a name for such an operation? In some sense, it is similar to taking the powerset of a set.

Thanks.

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