# A characterization of fine words over a finite alphabet

Glen, A. (2008) *A characterization of fine words over a finite alphabet.* Theoretical Computer Science, 391 (1-2). pp. 51-60.

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## Abstract

To any infinite word t over a finite alphabet A we can associate two infinite words min (t) and max (t) such that any prefix of min (t) (resp. max (t)) is the lexicographically smallest (resp. greatest) amongst the factors of t of the same length. We say that an infinite word t over A is fine if there exists an infinite word s such that, for any lexicographic order, min (t) = a s where a = min (A). In this paper, we characterize fine words; specifically, we prove that an infinite word t is fine if and only if t is either a strict episturmian word or a strict "skew episturmian word". This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.

Publication Type: | Journal Article |
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Publisher: | Elsevier BV |

Copyright: | © 2007 Elsevier Ltd. |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/3878 |

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