We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSO-interpretability in algebraic trees, and the decidability of the MSO theory of morphic words.Refining their techniques we observe that the lexicographicpresentation of a (morphic) word is in a certain sense canonical.We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties.We introduce k-lexicographic presentations, and morphisms oflevel k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic orlevel k morphic words. We prove that these presentations arealso canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classesunder MSO-definable recolorings, e.g. closure under deterministic sequential mappings.The classes of k-lexicographic words are shown to constitute an infinite hierarchy.
