Let $\Sigma_n = \{0, 1, ... , n-1\}$. Suppose $L \subseteq$ $\Sigma^*_n$, and let

$\mathcal{B}(L) = \{ x \in L : x \text{ is the lexicographically largest among all strings of length } |x| \text{ in } L \}$. For example, $\mathcal{B}(\{0,1\}^*) = 1^*$ and $\mathcal{B}(\epsilon \cup 1(0 \cup 01)^*) = (10)^*(\epsilon \cup 1)$. Prove that if $L$ is regular, then so is $\mathcal{B}(L)$. Any hints? I was thinking of perfect shuffling $L$ with itself, and then defining a morphism to compare each pair of characters in a shuffle to determine which is the lexicographically greater one.