Synchronization of primitive automata

TL;DR

This paper establishes new conditions under which primitive automata are guaranteed to be synchronizing, focusing on the roles of defect 1 letters and rank 2 words in ensuring synchronizability.

Contribution

It introduces novel criteria linking primitivity with synchronization, expanding understanding of automaton behavior under specific structural conditions.

Findings

Primitivity combined with a letter of defect 1 guarantees synchronization.

Presence of a rank 2 word ensures synchronizability in primitive automata.

New conditions extend previous synchronization theories.

Abstract

We exhibit new conditions under which a primitive automaton is synchronizing. In particular, we show that the primitivity of an automaton forces its synchronizability whenever the automaton has either a letter of defect 1 or a word of rank 2.