Scattered Algebraic Linear Orderings

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. Moreover, using a result of Courcelle together with a Mezei-Wright type result, we can show that the algebraic words are exactly those that are isomorphic to the lexicographic ordering of a deterministic context-free language. Algebraic well-orderings have been shown to be those well-orderings whose order type is less than ωω . We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω .