Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Preface to this edition
- Chapter 1 Words
- Chapter 2 Square-Free Words and Idempotent Semigroups
- Chapter 3 Van der Waerden's Theorem
- Chapter 4 Repetitive Mappings and Morphisms
- Chapter 5 Factorizations of Free Monoids
- Chapter 6 Subwords
- Chapter 7 Unavoidable Regularities in Words and Algebras with Polynomial Identities
- Chapter 8 The Critical Factorization Theorem
- Chapter 9 Equations in Words
- Chapter 10 Rearrangements of Words
- Chapter 11 Words and Trees
- Bibliography
- Index
Chapter 4 - Repetitive Mappings and Morphisms
Published online by Cambridge University Press: 04 November 2009
- Frontmatter
- Contents
- Foreword
- Preface
- Preface to this edition
- Chapter 1 Words
- Chapter 2 Square-Free Words and Idempotent Semigroups
- Chapter 3 Van der Waerden's Theorem
- Chapter 4 Repetitive Mappings and Morphisms
- Chapter 5 Factorizations of Free Monoids
- Chapter 6 Subwords
- Chapter 7 Unavoidable Regularities in Words and Algebras with Polynomial Identities
- Chapter 8 The Critical Factorization Theorem
- Chapter 9 Equations in Words
- Chapter 10 Rearrangements of Words
- Chapter 11 Words and Trees
- Bibliography
- Index
Summary
Introduction
This chapter is devoted to the study of a special type of unavoidable regularities. We consider a mapping φ:A+ → E from A+ to a set E, and we search in a word w for factors of the type w1w2 … wn with φ(w1) = φ(w2)= … = φ(wn). The mapping is called repetitive when such a factor appears in each sufficiently long word. This is related both to square-free words (Chapter 2), by considering the identity mapping, and to van der Waerden's theorem (Chapter 3), as will be shown later on.
It will first be shown that any mapping from A+ to a finite set is repetitive (Theorem 4.1.1).
After a direct proof of this fact, it will be shown how the result can also be deduced from Ramsey's theorem (which is stated without proof).
Investigated also is the special case where φ is a morphism from A+ to a semigroup S. First it is proved that a morphism to the semigroup of positive integers is repetitive when the alphabet is finite (Theorem 4.2.1). Then it is proved that a morphism to a finite semigroup is uniformly repetitive, in the sense that the words w1, w2,…, wn/i> in the foregoing definition can be chosen of equal length (Theorem 4.2.2). This is, as will be shown, a generalization of van der Waerden's theorem. Finally, the chapter mentions a number of extensions and other results.
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- Information
- Combinatorics on Words , pp. 55 - 62Publisher: Cambridge University PressPrint publication year: 1997