Reset thresholds of transformation monoids

TL;DR

This paper explores reset thresholds in transformation monoids, introducing monoidal automata to formulate the cernfd conjecture for monoids, and establishes upper bounds under specific structural conditions.

Contribution

It introduces monoidal automata to extend the cernfd conjecture to monoids and provides new upper bounds on reset thresholds for certain classes of monoids.

Findings

Quadratic upper bound on reset threshold for monoids with a primitive permutation group and a singular element of maximal rank.

Formulation of the cernfd conjecture within the framework of monoids.

Upper bounds depend on structural properties of the monoid.

Abstract

Motivated by the \v{C}ern\'y conjecture for automata, we introduce the concept of monoidal automata, which allows the formulation of the \v{C}ern\'y conjecture for monoids. We show upper bounds on the reset threshold of monoids with certain properties. In particular, we obtain a quadratic upper bound if the transformation monoid contains a primitive group of permutations and a singular of maximal rank with only one point of contraction.