Generating Sets of the Kernel Graph and the Inverse Problem in Synchronization Theory

Abstract

This paper analyses the construction of the kernel graph of a non-synchronizing transformation semigroup and introduces the inverse synchronization problem. Given a transformation semigroup $S\leq T_n$, we construct the kernel graph $\text{Gr}(S)$ by saying $v$ and $w$ are adjacent, if there is no $f\in S$ with $vf=wf$. The kernel graph is trivial or complete if the semigroup is a synchronizing semigroup or a permutation group, respectively. The connection between graphs and synchronizing (semi-) groups was established by Cameron and Kazanidis, and it has led to many results regarding the classification of synchronizing permutation groups, and the description of singular endomorphims of graphs. This paper, firstly, emphasises the importance of this construction mainly by proving its superior structure, secondly, analyses the construction and discusses minimal generating sets and their combinatorial properties, and thirdly, introduces the inverse synchronization problem. The third part also includes an additional characterization of primitive groups.