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 lexicographic order $slex$ 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 $slex$-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 cite{BE,BEa} to give a proof that the least nonregular ordinal is $omega^omega $.
An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wri
ght type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelles characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than $omega^omega$. It follows that the algebraic ordinals are exactly those less than $omega^{omega^omega}$.
