Reset Thresholds of Transformation Monoids

Motivated by the Čherný conjecture for automata, we introduce the concept of monoidal automata, which allows us to formulate the Čherný conjecture for monoids. We obtain the 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.