Completely reducible sets

TL;DR

This paper investigates a class of rational word sets called completely reducible sets, exploring their algebraic properties, closure characteristics, and generalizations of known results in formal language theory.

Contribution

It introduces new results on the linear representations of monoids related to completely reducible sets and extends existing theorems to birecurrent and cyclic sets.

Findings

Generalization of complete reducibility to birecurrent sets

New proof of cyclic sets' properties

Closure properties of completely reducible sets

Abstract

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.