The special subgroup of invertible non-commutative rational power series as a metric group

Roland Bacher (1)

1 IF - Institut Fourier

We give an easy proof of Schützenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group.

Mots clés: non-commutative formal power series; rational series; recognisable series; metric group; automaton sequence; regular language

Domaines: Combinatoire [math.CO]; Théorie des groupes [math.GR]

Roland Bacher. The special subgroup of invertible non-commutative rational power series as a metric group. 2008. ⟨hal-00270911v2⟩