By Jean Berstel, Dominique Perrin, Christophe Reutenauer

This significant revision of Berstel and Perrin's vintage thought of Codes has been rewritten with a extra smooth concentration and a much wider assurance of the topic. the idea that of unambiguous automata, that is in detail associated with that of codes, now performs an important function through the booklet, reflecting advancements of the final two decades. this can be 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 remedy is still self-contained: there's heritage fabric on discrete arithmetic, algebra and theoretical computing device technological know-how. The wealth of routines and examples make it perfect for self-study or classes. In precis, it is a entire reference at the concept of variable-length codes and their relation to automata.

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Proof. Let m ∈ H(e). Then, we have for some u, u′ , v, v ′ ∈ M e = mu , m = eu′ , e = vm , m = v′ e . Therefore em = e(eu′ ) = eu′ = m and in the same way me = m. This shows that m ∈ eM e. Since m(eue) = mue = e , (eve)m = evm = e , 1032 1033 1034 the element m is both right and left invertible in M . Hence, m belongs to the group of units of eM e. Conversely, if m ∈ eM e is right and left invertible, we have mu = vm = e for some u, v ∈ eM e. Since m = em = me, we obtain mHe. J. Berstel, D. Perrin and C.

If P = Q, we say that it is a K-relation over Q. The set of all K-relations between P and Q is denoted by K P ×Q . Let m ∈ K P ×Q be a K-relation between P and Q. For p ∈ P , the row of index p of m is denoted by mp∗ . It is the element of K Q defined by (mp∗ )q = mpq . Similarly, the column of index q of m is denoted by m∗q . It is an element of K P . Let P, Q, R be three sets and let K be a complete semiring. For m ∈ K P ×Q and n ∈ K Q×R , the product mn is defined as the following element of K P ×R .

Proof. Set N (z) = I − M z, where I is the identity matrix and z is a variable. The polynomial N (z) can be considered both as a polynomial with coefficients in the ring of m × m-matrices or as an m × m-matrix with coefficients in the ring of real polynomials in the variable z. The polynomial N (z) is invertible in both structures, and its inverse N (z)−1 = (I − M z)−1 can in turn be viewed as a power series with coefficients in the ring of m × m-matrices or as a matrix whose coefficients are rational fractions in the variable z.