Complementation of rational sets on scattered linear orderings of finite rank

In a preceding paper [V. Bruyère, O. Carton, Automata on linear orderings, in: J. Sgall, A. Pultr, P. Kolman (Eds.), MFCS’2001, in: Lect. Notes in Comput. Sci., vol. 2136, 2001, pp. 236–247. iGM report 2001-12], automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite, and even transfinite words studied by Büchi. Kleene’s theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation.