Groups Synchronizing a Transformation of Non-Uniform Kernel

TL;DR

This paper explores the classification of finite automata with a focus on the properties of their transition semigroups, introducing the concept of almost synchronizing groups and providing new examples of such groups.

Contribution

It introduces the notion of almost synchronizing groups and presents infinite families of groups that are not fully synchronizing but are almost so, advancing the understanding of automata synchronization.

Findings

Identified infinite families of almost synchronizing groups.

Demonstrated the existence of groups that are not synchronizing but almost so.

Proposed open problems in synchronization theory.

Abstract

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure. Let $X$ be a finite set. We say that a primitive group $G$ on $X$ is {\em synchronizing} if $G$ together with any non-invertible map on $X$ generates a constant map. It is known (by some recent results proved by P. M. Neumann) that for some primitive groups $G$ and for some singular transformations $t$ of uniform kernel (that is, all blocks have the same number of elements), the semigroup $< G,t>$ does not generate a constant map. Therefore the following concept is very natural: a primitive group $G$ on $X$ is said to be {\em almost synchronizing} if $G$ together with any map of non-uniform kernel generates a constant map. In this paper we use two different methods to provide several infinite families of groups that are not synchronizing, but are almost synchronizing. The paper ends with a number of problems on synchronization likely to attract the attention of experts in computer science, combinatorics and geometry, groups and semigroups, linear algebra and matrix theory.