On automatic infinite permutations∗
Sobolev Institute of Mathematics SB RAS, Koptyug av. 4,
2 Institut Camille Jordan, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.
3 Department of Mathematics and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland.
Accepted: 10 October 2011
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.
Mathematics Subject Classification: 05A05 / 68R15
Key words: Permutation / infinite permutation / ordering / infinite word / automatic word / automatic permutation / Thue-Morse word / Thue-Morse permutation
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