By Jean Berstel
The algebraic conception of automata was once created by way of Sch?tzenberger and Chomsky over 50 years in the past and there has for the reason that been loads of improvement. Classical paintings at the concept to noncommutative strength sequence has been augmented extra lately to parts similar to illustration concept, combinatorial arithmetic and theoretical machine technology. This booklet provides to an viewers of graduate scholars and researchers a latest account of the topic and its purposes. The algebraic procedure permits the speculation to be built in a common type of large applicability. for instance, number-theoretic effects can now be extra absolutely explored, as well as functions in automata thought, codes and non-commutative algebra. a lot fabric, for instance, Sch?tzenberger's theorem on polynomially bounded rational sequence, appears to be like the following for the 1st time in e-book shape. this is often a great source and reference for all these operating in algebra, theoretical machine technology and their components of overlap.
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Additional resources for Noncommutative Rational Series with Applications (Encyclopedia of Mathematics and its Applications)
This implies in turn that µ(K A ) is a finitely generated K-module. Now Ker(µ) is an ideal contained in Ker(S). Thus by definition Ker(µ) ⊂ IS , and MS is a quotient of µ(K A ). Hence it is a finitely generated module over K. Conversely, suppose that the syntactic algebra of S is a finitely generated module over K. Consider, for each word w in A∗ , the K-endomorphism νw of MS defined by m → νw(m) = µS (w)m . 1. S YNTACTIC IDEALS 29 The function ν : A∗ → End(MS ) is a monoid morphism, and moreover (S, w) = φS ◦ µS (w) = φS (µS (w)) = φS (νw(1)) .
Show that p ◦ i : K → L is injective if and only if for all a, b, c ∈ K a + b = a + c =⇒ b = c . 464 465 A semiring having this property is called regular. Show that K can be embedded into a ring if and only if it is regular. Exercises for Chapter 1 21 c) Assume that K is regular. Show that the ring L is without zero divisors if and only if for all a, b, c, d ∈ K, the following condition holds: ac + bd = ad + bc =⇒ a = b or c = d . Show that K can be embedded into a field if and only if K is regular and this condition is satisfied.
For any word w over A, we denote by νk (w) the integer represented by w in base k. For example νk (0111) = k 2 + k + 1. We write c for c when we need to distinguish the symbol c from the number c. Let S and T be the series defined by S= k |w| w . νk (w) w, T = w w Exercises for Chapter 1 23 Show that T = 1 + kAT and that S = P T + AS. Deduce that S = A∗ P (kA)∗ , 514 515 516 517 where P = 1 + 22 + · · · (k − 1)k − 1. 5 Assume that K is a ring. Show that a series is invertible in K A if and only if its constant term is invertible in K.
Noncommutative Rational Series with Applications (Encyclopedia of Mathematics and its Applications) by Jean Berstel