A Note on Ordinal DFAs

We prove the following theorem. Suppose that M is a trim DFA on the Boolean alphabet 0, 1. The language L(M) is well-ordered by the lex-icographic order < ℓ iff whenever the non sink states q, q.0 are in the same strong component, then q.1 is a sink. It is easy to see that this property is sufficient. In order to show the necessity, we analyze the behavior of a < ℓ-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA M , whether L(M) is well-ordered by the lexicographic order. Last, we apply an argument in [BE10, BE10a] to give a proof that the least nonregular ordinal is ω ω .