Simplicity and irreducibility in circular automata

TL;DR

This paper characterizes when circular automata are simple or irreducible, introducing a new perspective on irreducibility over complex numbers and providing criteria based on the weak contracting property.

Contribution

It offers a complete characterization of simplicity and irreducibility in circular automata using the weak contracting property and extends classical results to complex monoid representations.

Findings

Complete characterization of simplicity in circular automata.

Necessary and sufficient conditions for irreducibility of contracting automata.

Examples illustrating the theoretical results.

Abstract

This paper investigates the conditions under which a given circular (synchronizing) DFA is \emph{simple} (sometimes referred to as \emph{primitive}) and when it is \emph{irreducible}. Our notion of irreducibility slightly differs from the classical one, since we are considering our monoid representations to be over $\mathbb{C}$ instead of $\mathbb{Q}$; nevertheless, several well-known results remain valid-for instance, the fact that every irreducible automaton is necessarily simple. We provide a complete characterization of simplicity in the circular case by means of the \emph{weak contracting property}. Furthermore, we establish necessary and sufficient conditions for a circular \emph{contracting automaton} (a stronger condition than the weakly contracting one) to be irreducible, and we present examples illustrating our results.