Séries formelles et algèbres syntactiques

Abstract

The notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleene's theorem, Schützenberger's theorem about aperiodic monoid and Eilenberg's theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schützenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schützenberger's theorem on aperiodic monoids.