The hereditariness problem for the Černý conjecture
Emanuele Rodaro, Riccardo Venturi
TL;DR
This work tackles the hereditariness problem for the Černý conjecture by embedding the problem into a representation-theoretic framework that relates congruences of a synchronizing automaton to ideals in its transition monoid via a Galois connection. By introducing and analyzing radical, simple, and quasi-simple automata, the authors show that proving the Černý bound for these three classes suffices to settle the conjecture in general, with extremal examples confined to them. They establish a structural correspondence between the radical ideal and radical congruence, bound the nilpotency index of the radical by the height of the congruence lattice, and develop algorithms to compute radicals. The main result reduces the broad hereditariness question to three well-structured classes, providing a clear pathway to progress on the Černý conjecture and linking it to Wedderburn–Artin type decompositions in the associated synchronized algebras. The work thus bridges automata theory with algebraic representation theory to offer new reduction principles and computational tools for reset words.
Abstract
This paper addresses the lifting problem for the Černý 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 Černý 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.