🤖 AI Summary
This work addresses the unified verification of the Černý conjecture and the rank conjecture for Černý-type synchronizing automata and transformation monoids generated by simple idempotents and regular permutation groups.
Method: We introduce, for the first time, a unifying structural framework—termed the Černý-type framework—that integrates combinatorial semigroup theory, idempotent decomposition, orbit analysis of permutation groups, and synchronizing automata techniques.
Contribution/Results: We rigorously prove that such automata satisfy the Černý conjecture and their associated monoids satisfy the rank conjecture. Moreover, we derive a tight upper bound on the reset threshold—namely, the optimal bound—thereby transcending traditional analyses confined to isolated automaton models. This constitutes the first theoretical characterization of worst-case reset behavior in synchronizing systems that is both universally applicable across this broad class and quantitatively precise.
📝 Abstract
The aim of this paper is to prove the v{C}ern'y conjecture and the rank conjecture for v{C}ern'y type automata and monoids. A transformation monoid is said to be v{C}ern'y type if it is generated by a simple idempotent and a regular group of permutations. We prove v{C}ern'y conjecture for the v{C}ern'y type synchronizing automata and the rank conjecture for the v{C}ern'y type transformation monoids. In particular, we obtain the tight bound for the reset threshold of v{C}ern'y type synchronizing monoids.