Cerny type automata and rank conjecture

TL;DR

This paper proves the Cerny and rank conjectures for Cerny type automata and monoids, establishing tight bounds for their reset thresholds and advancing understanding of their synchronization properties.

Contribution

It introduces the concept of Cerny type automata and monoids and proves the Cerny and rank conjectures specifically for these classes, including tight bounds.

Findings

Proved Cerny conjecture for Cerny type automata

Proved rank conjecture for Cerny type monoids

Established tight bounds for reset thresholds

Abstract

The aim of this paper is to prove the \v{C}ern\'y conjecture and the rank conjecture for \v{C}ern\'y type automata and monoids. A transformation monoid is said to be \v{C}ern\'y type if it is generated by a simple idempotent and a regular group of permutations. We prove \v{C}ern\'y conjecture for the \v{C}ern\'y type synchronizing automata and the rank conjecture for the \v{C}ern\'y type transformation monoids. In particular, we obtain the tight bound for the reset threshold of \v{C}ern\'y type synchronizing monoids.