doi:10.1016/j.tcs.2005.03.037 
Copyright © 2005 Elsevier B.V. All rights reserved.
aDepartment of Mathematics, University of Turku and Turku Centre for Computer Science, Turku 20014, Finland
bLIFL, URA CNRS 369, Université des Sciences et Technologie de Lille, F-59655 Villeneuve d’Ascq, France
cDepartment of Computer Science, Åbo Akademi University and Turku Centre for Computer Science, Turku 20520, Finland
Available online 16 April 2005.
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Abstract
The centralizer of a set of words X is the largest set of words 
commuting with X: 
. It has been a long standing open question due to [J.H. Conway, Regular Algebra and Finite Machines, Chapman & Hall, London (1971).], whether the centralizer of any rational set is rational. While the answer turned out to be negative in general, see [M. Kunc, Proc. of ICALP 2004, Lecture Notes in Computer Science, Vol. 3142, Springer, Berlin, 2004, pp. 870–881.], we prove here that the situation is different for codes: the centralizer of any rational code is rational and if the code is finite, then the centralizer is finitely generated. This result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by [B. Ratoandromanana, RAIRO Inform. Theor. 23(4) (1989) 425–444.]—many of the techniques we use in this paper follow her ideas. We also give in this paper an elementary proof for the prefix case.
Keywords: Codes; Commutation; Centralizer; Conway's problem; Prefix codes
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2, and in particular, we study the properties of codes with degree 2.
