Missing factors of ideals and synchronizing automata

Recently, a series of papers have started to look at Cerný‘s conjecture, and in general at synchronizing automata, from the point of view of the theory of ideals of free monoids. The starting point of such an approach is a simple observation: the set of reset words of an automaton is a two-sided ideal of the free monoid on its alphabet that is also a regular language. We study the relationship between a synchronizing automaton and the sets of (minimal) generators of its reset words. We show that if such set does not contain a word of a certain length, then Cerný‘s conjecture holds.