  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.
