Noncommutative Rational Series with ApplicationsThe algebraic theory of automata was created by Schützenberger and Chomsky over 50 years ago and there has since been a great deal of development. Classical work on the theory to noncommutative power series has been augmented more recently to areas such as representation theory, combinatorial mathematics and theoretical computer science. This book presents to an audience of graduate students and researchers a modern account of the subject and its applications. The algebraic approach allows the theory to be developed in a general form of wide applicability. For example, number-theoretic results can now be more fully explored, in addition to applications in automata theory, codes and non-commutative algebra. Much material, for example, Schützenberger's theorem on polynomially bounded rational series, appears here for the first time in book form. This is an excellent resource and reference for all those working in algebra, theoretical computer science and their areas of overlap. |
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0-minimal a₁ assume automata automaton bijection Chapter characteristic series commutative ring complete congruence Consequently consider constant term contains Corollary cycle complexity defined definition deg(P deg(Y denoted eigenvalues element equal Equation Exercise exist polynomials Fatou field follows formal series free monoid graph Hankel matrix height function Hence idempotent identity implies induction integer invertible K-linear K-module k-regular K-submodule Knxn Lemma linear combination linear recurrence relation linear representation M₁ matrix minimal linear representation minimal polynomial module Moreover morphism N-rational noncommutative nonzero obtain P₁ Proof of Theorem Proposition prove rational expression rational language rational series recurrence relation resp Reutenauer right ideal S₁ satisfies Schützenberger Section semigroup semiring semisimple sequence Show Soittola stable star height strongly connected component submodule subset subspace supp(S Suppose syntactic algebra syntactic ideal syntactic monoid Theorem 1.1 V₁ vector word αΕΑ