The van der Waerden theorem and the Burnside problem for semigroups (Q1113288)

The well known van der Waerden Theorem on arithmetic progressions is a particular case [see the author in M. Lothaire, Combinatorics on words (1983; Zbl 0514.20045)] of the following statement: any finite semigroup is uniformly repetitive, i.e. for every finite alphabet A, for every morphism \(\phi\) from the free semigroup \(A^+\) into the semigroup S and for every positive integer k there exists an integer \(h=h(\phi,k)\) such that every word w of \(A^+\) having at least length h contains k consecutive factors having the same positive length and the same image under \(\phi\). Now, trying to give an answer to a natural question (are the finitely generated and uniformly repetitive semigroups finite?) we prove the main result of this paper: any uniformly repetitive semigroup is periodic. - A result concerning the Burnside problem for semigroups is also given.