By Jean Berstel, Dominique Perrin, Christophe Reutenauer

This significant revision of Berstel and Perrin's vintage idea of Codes has been rewritten with a extra glossy concentration and a much wider assurance of the topic. the concept that of unambiguous automata, that's in detail associated with that of codes, now performs an important position during the publication, reflecting advancements of the final two decades. this is often complemented by way of a dialogue of the relationship among codes and automata, and new fabric from the sector of symbolic dynamics. The authors have additionally explored hyperlinks with more effective functions, together with information compression and cryptography. The therapy is still self-contained: there's historical past fabric on discrete arithmetic, algebra and theoretical machine technological know-how. The wealth of routines and examples make it excellent for self-study or classes. In precis, it is a complete reference at the concept of variable-length codes and their relation to automata.

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**Example text**

Let m, m′ ∈ M be R-equivalent. Let u, u′ ∈ M be such that m = m ′ u′ , m′ = mu . The mappings ρu : q → qu, 1016 ρu′ : q ′ → q ′ u′ are bijections from L(m) onto L(m′ ) inverse to each other which map an R-class onto itself. Version 14 janvier 2009 J. Berstel, D. Perrin and C. Reutenauer fig0_06 42 1017 1018 1019 1. P RELIMINARIES Proof. We first verify that ρu maps L(m) into L(m′ ). If q ∈ L(m), then M q = M m and therefore M qu = M mu = M m′ . Hence qu = ρu (q) is in L(m′ ). Analogously, ρu′ maps L(m′ ) into L(m).

Let m, n ∈ I \ 0. By 2, the right ideal mM and the left ideal M n are 0-minimal. Since M is prime, there exists u ∈ M such that mun = 0. The right ideal mM being 0-minimal, we have mM = munM and therefore mRmun. In the same way, munLn. It follows that mDn. This shows that I \ 0 is contained in a D-class. Conversely, if m ∈ I \ 0, n ∈ M and mDn, there exists a k ∈ M such that mM = kM and M k = M n. Consequently I = M mM = M kM = M nM and this implies n ∈ I \ 0. 7 Let us show that I \0 is a regular D-class.

4 (σi , w) . 20) all but a finite number of terms are different from 0. We easily check that for a locally finite family (σi )i∈I of elements of K A and any τ in K A , we have σi = τ i∈I τ σi . i∈I Let σ ∈ K A be a series. The constant term of σ is the element (σ, 1) of K. If σ has zero constant term, then the family (σ n )n≥0 is locally finite, because the support of σ n does not contain words of length less than n. We denote by σ ∗ and by σ + the series σ∗ = σn , n≥0 692 σ+ = σn . n≥1 The series σ ∗ is called star of σ.