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

TL;DR

This paper explores the kernel graph construction in transformation semigroups, analyzes its structure and properties, and introduces the inverse synchronization problem, advancing understanding of synchronization theory and group classifications.

Contribution

It provides a detailed analysis of the kernel graph construction, discusses minimal generating sets, and introduces the inverse synchronization problem in the context of synchronization theory.

Findings

Kernel graph construction has a rich structure with implications for synchronization.

Minimal generating sets exhibit notable combinatorial properties.

The inverse synchronization problem offers new insights into group classifications.

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.