Uniform winning strategies for the synchronization games on subclasses of finite automata

TL;DR

This paper demonstrates a uniform winning strategy for the synchronization game on automata with transition monoids in the pseudovariety DS, establishing it as the largest such class with this property.

Contribution

It introduces a uniform winning strategy for a class of automata and characterizes the maximal pseudovariety with this property.

Findings

A uniform winning strategy exists for automata with transition monoids in DS.

S is the largest pseudovariety with this synchronization property.

Abstract

The pseudovariety $\mathbf{DS}$ consists of all finite monoids whose regular $D$-classes form subsemigroups. We exhibit a uniform winning strategy for Synchronizer in the synchronization game on every synchronizing automaton whose transition monoid lies in $\mathbf{DS}$, and we prove that $\mathbf{DS}$ is the largest pseudovariety with this property.