# Birecurrent sets

## TL;DR

This paper investigates birecurrent sets, which are recurrent sets with strongly connected minimal automata and their reversals, providing characterizations and properties related to their reducibility and structure.

## Contribution

It characterizes completely reducible sets as linear combinations of birecurrent sets, advancing understanding of their algebraic and automata-theoretic properties.

## Key findings

- Birecurrent sets are completely reducible.
- Main result: completely reducible sets are linear combinations of birecurrent sets.
- Provides new characterizations of birecurrent sets.

## Abstract

A set is called recurrent if its minimal automaton is strongly connected and birecurrent if it is recurrent as well as its reversal. We prove a series of results concerning birecurrent sets. It is already known that any birecurrent set is completely reducible (that is, such that the minimal representation of its characteristic series is completely reducible). The main result of this paper characterizes completely reducible sets as linear combinations of birecurrent sets