The hereditariness problem for the \v{C}ern\'y conjecture

Abstract

This paper addresses the lifting problem for the \v{C}ern\'y conjecture: namely, whether the validity of the conjecture for a quotient automaton can always be transferred (or "lifted") to the original automaton. Although a complete solution remains open, we show that it is sufficient to verify the \v{C}ern\'y conjecture for three specific subclasses of reset automata: radical, simple, and quasi-simple. Our approach relies on establishing a Galois connection between the lattices of congruences and ideals of the transition monoid. This connection not only serves as the main tool in our proofs but also provides a systematic method for computing the radical ideal and for deriving structural insights about these classes. In particular, we show that for every simple or quasi-simple automaton $\mathcal{A}$, the transition monoid $\text{M}(\mathcal{A})$ possesses a unique ideal covering the minimal ideal of constant (reset) maps; a result of similar flavor holds for the class of radical automata.