Primitive Automata that are Synchronizing

TL;DR

This paper investigates the relationship between primitive and synchronizing properties in deterministic automata, providing partial characterizations and proposing conjectures for broader classes.

Contribution

It proves that certain classes of primitive automata are necessarily synchronizing and introduces conjectures for more general cases.

Findings

Primitive automata with permutation or semiconstant transformations are synchronizing.

The paper establishes implications for specific automata classes.

Proposes conjectures for characterizing primitive automata that are synchronizing.

Abstract

A deterministic finite (semi)automaton is primitive if its transition monoid (semigroup) acting on the set of states has no non-trivial congruences. It is synchronizing if it contains a constant map (transformation). In analogy to synchronizing groups, we study the possibility of characterizing automata that are synchronizing if primitive. We prove that the implication holds for several classes of automata. In particular, we show it for automata whose every letter induce either a permutation or a semiconstant transformation (an idempotent with one point of contraction) unless all letters are of the first type. We propose and discuss two conjectures about possible more general characterizations.