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 $\text{L}(M)$ is well-ordered by the lexicographic order $\text{slex}$ 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 $\text{slex}$-descending sequence of words. This property is used to obtain a polynomial time algorithm to determine, given a DFA $M$, whether $\text{L}(M)$ is well-ordered by the lexicographic order. Last, we apply an argument in \cite{BE,BEa} to give a proof that the least nonregular ordinal is ${\omega }^{\omega }$.