Ideal regular languages and strongly connected synchronizing automata

We introduce the notion of reset left regular decomposition of an ideal regular language and we prove that the category formed by these decompositions with certain morphisms is equivalent to the category of strongly connected synchronizing automata. We show that each ideal regular language has at least a reset left regular decomposition. As a consequence, each ideal regular language is the set of synchronizing words of some strongly connected synchronizing automaton. Furthermore, this one-to-one correspondence allows us to introduce the notion of reset decomposition complexity of an ideal. This notion allows the reformulation of Černý’s conjecture in pure language theoretic terms.