Reset thresholds of transformation monoids

Motivated by the Černý conjecture for automata, we introduce the concept of monoidal automata, which allows the formulation of the Černý 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.