Algebraic Characterization of FO for Scattered Linear Orderings

We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked product, and recognizability by finite aperiodic semigroups which satisfy the equation xωx−ω = x. 1998 ACM Subject Classification F.4.1 Mathematical Logic; F.4.3 Formal Languages