Dixon's Theorem and random synchronization

TL;DR

This paper investigates the probability that a randomly generated submonoid of the full transformation monoid is synchronizing, extending Dixon's theorem to transformation monoids and linking them to graph endomorphisms.

Contribution

It develops a novel connection between transformation monoids and graph endomorphisms to analyze synchronization properties.

Findings

Established a link between non-synchronizing monoids and graph endomorphisms.

Provided a framework to analyze the probability of synchronization in random transformation monoids.

Extended Dixon's theorem concepts to the context of transformation monoids.

Abstract

A transformation monoid on a set Omega is called synchronizing if it contains an element of rank 1 (that is, mapping the whole of Omega to a single point). In this paper, I tackle the question: given n and k, what is the probability that the submonoid of the full transformation monoid T_n generated by k random transformations is synchronizing? This question is analogous to Dixon's Theorem that two random permutations generate the symmetric or alternating group with high probability. Following the technique of Dixon's theorem, we need to analyse the maximal non-synchronizing submonoids of T_n. I develop a very close connection between transformation monoids and graphs, from which we obtain a description of non-synchronizing monoids as endomorphism monoids of graphs satisfying some very strong conditions. However, counting such graphs, and dealing with the intersections of their endomorphism monoids, seems difficult.