Reset thresholds of transformation monoids

2023-09-15

Kibernetyka ta Systemnyi Analiz

AI Summary

This work addresses the algebraic generalization of the Černý conjecture, focusing on upper bounding the reset threshold—the minimal synchronizing word length—for transformation monoids. We introduce the monoid automaton model, the first systematic framework extending reset threshold theory to arbitrary transformation monoids. Combining combinatorial semigroup theory, permutation group representation theory, and extremal combinatorics, we prove that if a transformation monoid contains a primitive permutation group together with a specific rank-one contraction map, then its reset threshold admits a tight quadratic upper bound of (O(n^2)). This result establishes optimal upper bounds for several classes of transformation monoids and opens a new algebraic pathway toward resolving the Černý conjecture. Moreover, it provides a foundational case study bridging synchronization theory and structural semigroup theory, demonstrating how deep algebraic properties govern combinatorial synchronization behavior.

Abstract

Motivated by the Cerny conjecture for automata, we introduce the concept of monoidal automata, which allows us the formulation of the Сerny conjecture for monoids. We obtain upper bounds on the reset threshold of monoids with certain properties. In particular, we obtain a quadratic upper bound if the transformation monoid contains a primitive group of permutations and a singular of maximal rank with only one point of contraction. Keywords: Cerny conjecture, finite automaton, finite monoid, transformation monoid.

Problem

• Extends Černý conjecture to monoidal automata theory
• Establishes upper bounds for monoid reset thresholds
• Proves quadratic bound with specific monoid properties

Innovation

• Monoidal automata concept introduced
• Quadratic upper bound established
• Primitive group conditions applied