Abstract
In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.
Highlights
Finite Lyndon words can be encountered in many studies
This result was extended to infinite words [21]
The study presented here aims to answer the natural questions: what can be said about morphisms preserving infinite Lyndon words or morphisms preserving the lexicographic order on infinite words? Do morphisms preserving infinite Lyndon words or morphisms preserving the lexicographic order on infinite words have testable characterizations? How are the families of these morphisms related to those of Lyndon morphisms and morphisms preserving the order on infinite words?
Summary
Finite (or infinite) Lyndon words can be encountered in many studies (see for instance [10, 11, 12]). To obtain the decomposition in Lyndon words of the (infinite) Thue-Morse word, Ido and Melancon [7] give some morphisms that preserve finite Lyndon words. Others characterizations provide algorithms to test if a morphism preserves Lyndon words or the order on finite words. The study presented here aims to answer the natural questions: what can be said about morphisms preserving infinite Lyndon words or morphisms preserving (possibly strictly) the lexicographic order on infinite words? We show that they are exactly the morphisms preserving finite Lyndon words of length n0 for any given integer n0 ≥ 2. An extended abstract of this paper was presented during the Journees Montoises 2004 [19]
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