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**Algebraic combinatorics on words.**
*(English)*
Zbl 1001.68093

Encyclopedia of Mathematics and Its Applications. 90. Cambridge: Cambridge University Press. xiii, 504 p. £60.00/hbk (2002).

This book is a continuation of Lothaire’s first book entitled “Combinatorics on words” (1983; Zbl 0514.20045), but it is independent. Compared to the previous one, this book both gives new topics and goes deeper into topics already present before.

The first chapter serves as an introduction. The second one is devoted to Sturmian words and Sturmian morphisms, and to their relation to continued fractions. The third chapter concentrates on unavoidable patterns in infinite words. Chapter 4 studies sesquipowers. Chapter 5 addresses the plactic monoid. The sixth chapter is devoted to codes (in the sense of Schützenberger). The seventh one studies numeration systems. Chapter 8 deals with periodicity. Chapter 9 addresses the question of centralizers of non-commutative series and polynomials. Chapter 10 studies transformations on words and \(q\)-calculus. Chapter 11 is devoted to statistics on permutations and words. Chapter 12 deals with Makanin’s algorithm. Chapter 13 studies independent systems of equations in semigroups.

This book will certainly become a reference book and have the same impact as the first book of Lothaire: essentially self-contained, with many exercises and interesting notes, not mentioning a bibliography with more than 450 items. Finally, note that “Lothaire” is as in the first book – a collective name, but that the authors are not exactly the same.

The first chapter serves as an introduction. The second one is devoted to Sturmian words and Sturmian morphisms, and to their relation to continued fractions. The third chapter concentrates on unavoidable patterns in infinite words. Chapter 4 studies sesquipowers. Chapter 5 addresses the plactic monoid. The sixth chapter is devoted to codes (in the sense of Schützenberger). The seventh one studies numeration systems. Chapter 8 deals with periodicity. Chapter 9 addresses the question of centralizers of non-commutative series and polynomials. Chapter 10 studies transformations on words and \(q\)-calculus. Chapter 11 is devoted to statistics on permutations and words. Chapter 12 deals with Makanin’s algorithm. Chapter 13 studies independent systems of equations in semigroups.

This book will certainly become a reference book and have the same impact as the first book of Lothaire: essentially self-contained, with many exercises and interesting notes, not mentioning a bibliography with more than 450 items. Finally, note that “Lothaire” is as in the first book – a collective name, but that the authors are not exactly the same.

Reviewer: Jean-Paul Allouche (Orsay)

### MSC:

68R15 | Combinatorics on words |

68-02 | Research exposition (monographs, survey articles) pertaining to computer science |

11B85 | Automata sequences |

### Citations:

Zbl 0514.20045
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\textit{M. Lothaire}, Algebraic combinatorics on words. Cambridge: Cambridge University Press (2002; Zbl 1001.68093)

### Online Encyclopedia of Integer Sequences:

Length of the n-th Zimin word (A082215(n)).Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value.

Result of the morphing 01->01021212, 02->0102121201, 12->01021201, iterated from ’01’. Sequence of the Fibonacci word fractal.

Fixed point of the morphism 0->011, 1->010.

Infinite Fibonacci word fractal sequence.

The Pell word: Fixed point of the morphism 0->001, 1->0.