RAIRO - Theoretical Informatics and Applications

Research Article

On some problems related to palindrome closure

Michelangelo Buccia1, Aldo de Lucaa1, Alessandro De Lucaa1 and Luca Q. Zambonia2

a1 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli Federico II. Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy; micbucci@unina.it; aldo.deluca@unina.it; alessandro.deluca@unina.it

a2 Department of Mathematics, PO Box 311430, University of North Texas. Denton TX, USA; luca@unt.edu

Abstract

In this paper, we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A*, then the right and left ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from a nonempty word. We show that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, we prove that for any given pseudostandard word s with seed, all sufficiently long left special factors of s are prefixes of it.

(Received January 9 2007)

(Accepted December 4 2007)

(Online publication January 15 2008)

Key Words:

  • Palindromes;
  • palindrome closures;
  • Sturmian and episturmian words;
  • involutory antimorphisms;
  • pseudopalindromes;
  • pseudostandard words.

Mathematics Subject Classification:

  • 68R15
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