The hereditariness problem for the Černý conjecture

2025-09-22

AI Summary

This study addresses the hereditary property of the Černý conjecture: whether its validity for a quotient automaton implies validity for the original automaton. To tackle this, we introduce three classes of reset automata—radical, simple, and quasi-simple—as a sufficient verification framework. Methodologically, we establish a Galois connection between the congruence lattice and the ideal lattice of the transition monoid, providing a novel algebraic paradigm for structural analysis. Building on this, we systematically characterize radical ideals and prove that the transition monoids of all three automaton classes possess a unique minimal constant-mapping ideal covering the entire monoid. These results unify key algebraic features across diverse automaton families and advance local-structural verification of the Černý conjecture. Crucially, they supply essential algebraic tools and a principled classification scheme toward resolving this longstanding open problem in combinatorial automata theory.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $ ext{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.

Problem

• Investigates whether the Černý conjecture transfers from quotient to original automata
• Determines sufficient conditions by verifying conjecture for three reset automata subclasses
• Establishes Galois connection between congruence lattices and transition monoid ideals

Innovation

• Galois connection between congruences and ideals
• Focus on radical, simple, quasi-simple automata
• Transition monoid's unique ideal covering minimal ideal