Noncommutative Rational Series with Applications by Jean Berstel

By Jean Berstel

The algebraic thought of automata was once created by way of Sch?tzenberger and Chomsky over 50 years in the past and there has because been loads of improvement. Classical paintings at the concept to noncommutative strength sequence has been augmented extra lately to components comparable to illustration concept, combinatorial arithmetic and theoretical desktop technology. This publication provides to an viewers of graduate scholars and researchers a latest account of the topic and its functions. The algebraic technique permits the idea to be constructed in a common kind of vast applicability. for instance, number-theoretic effects can now be extra absolutely explored, as well as purposes in automata conception, codes and non-commutative algebra. a lot fabric, for instance, Sch?tzenberger's theorem on polynomially bounded rational sequence, looks right here for the 1st time in ebook shape. this can be a superb source and reference for all these operating in algebra, theoretical computing device technology and their components of overlap.

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Next, the space S ◦ K A has as set of generators {S ◦ x | x ∈ A∗ }. Thus rank(S) is equal to the rank of this set. Since each S ◦ x can be identified with the row of index x in the Hankel matrix of S, the rank of S is equal to the rank of this matrix. If S is rational, it has a linear representation (λ, µ, γ) of dimension n. The right ideal J = {P ∈ K A | λµ(P ) = 0} 662 663 is contained in Ker(S), and its codimension is ≤ n. Consequently, J is contained in ISr , showing that rank(S) = codim(ISr ) ≤ codim(J) ≤ n.

2. S ERIES 45 AND RATIONAL LANGUAGES Proof. The following relations hold for series S and T in N A : supp(S + T ) = supp(S) ∪ supp(T ) , supp(ST ) = supp(S) supp(T ) , supp(S ∗ ) = (supp(S))∗ if S is proper. 976 977 978 979 980 981 982 It follows easily that the support of a rational series in N A is a rational language. For the converse, one can use the same relations, provided one has proved that any rational language can be obtained from finite sets by union, product, and submonoid generation restricted to proper languages (that is languages not containing the empty word).

It follows that S = λγ αa a a∈A ∗ = λγ 1 − αa a −1 . a∈A An example of a geometric series is the characteristic series of A∗ : S= w= w∈A∗ a a∈A ∗ = 1− a −1 . 3 The series S = w∈A∗ |w|a w has rank 2. 3). Next, the subdeterminant of its Hankel matrix corresponding to the rows and columns 1 and a is 0 1 = −1 . 1 2 669 Thus, S has rank ≥ 2. 6, the rank of S is 2. 670 2 Minimal linear representations 671 K denotes a field. 1 A linear representation (λ, µ, γ) of dimension n of a series S is minimal if and only if, setting M = µ(K A ), λM = K 1×n and Mγ = K n×1 .

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