An infinite permutation α is a linear ordering of N. We study properties
of infinite permutations analogous to those of infinite words, and show some resemblances
and some differences between permutations and words. In this paper, we try to extend to
permutations the notion of automaticity. As we shall show, the standard definitions which
are equivalent in the case of words are not equivalent in the context of permutations. We
investigate the relationships between these definitions and prove that they constitute a
chain of inclusions. We also construct and study an automaton generating the Thue-Morse
permutation.
