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

TL;DR

This paper provides a simplified proof of Schützenberger's Theorem linking rational and recognizable non-commutative power series, and introduces a natural metric on a subgroup of invertible rational series, exploring its properties.

Contribution

It offers an easy proof of a fundamental theorem and defines a new metric structure on a subgroup of invertible rational non-commutative power series.

Findings

Established a natural metric on the subgroup of invertible rational series

Described key features of this metric group

Simplified proof of Schützenberger's Theorem

Abstract

We give an easy proof of Sch\"utzenberger'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.