Context-free ordinals

We consider context-free languages equipped with the lexicographic ordering. We show that when the lexicographic ordering of a context-free language is scattered, then its Hausdorff rank is less than $ω^ω$. As a corollary of this result we obtain that an ordinal is the order type of a well-ordered context-free language iff it is less than $ω^{ω^ω}$.

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