  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-Wright type result, algebraic linear orderings are
exactly those isomorphic to the linear ordering of the leaves of an algebraic
tree. Using Courcelle's 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 $\omega^{\omega^\omega}$.
