By M. Lothaire

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For further references see Chapters 8 and 9. 4 is Eilenberg 1974. 1. (Levi's lemma). A monoid M is free iff there exists a morphism X of M into the monoid N of additive integers such that X~ \0) = 1M and if for any JC, y, z, /E M xy — zt implies the existence of a uE M such that either x = zw, uy -1 or xu — z, y — ut. 2. Let A be an alphabet and A = {a\a_E A] be a copy of A. Consider in the free monoid over the set A U A the congruence generated by the relations ad — da = 1, aEA. a. Show that each word has a unique representative of minimal length, called a reduced word.

3) for ao — a and also for ao = b. Consequently, iteration of //, on a and on b yields two infinite words By definition, t is the infinite word of Thue-Morse. {b) — baababba t = abbabaabbaababbabaababbaabbabaab••• t = baababbaabbabaababbabaabbaababba••• There are several properties relating the words \xn{a), iin(b), n^O. Consider the morphism defined by a — b, b—a Thus wjs obtained from w by replacing each a by b and conversely. Of course w = w. 1. Define uo = a, vo-b U n+\ = Then for U V n n> and for n V n+\ = V U n n' alln^O (i)un = ^(a), vH = f{b).

This work is of an essentially combinatorial nature. More recently, results from ergodic theory have led to the discovery of new extensions of van der Waerden's theorem, and, as a result, to a topological proof. The plan of the chapter illustrates this diversity of viewpoints. The first section, after a brief historical note, presents several different formulations of van der Waerden's theorem. The second section gives a combinatorial proof of an elegant generalization due to Griinwald. The third section, which concerns "cadences," gives an interpretation of the theorem in terms of the free monoid.