Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks

TL;DR

This paper advances the understanding of primitive groups' ability to synchronize non-uniform maps, especially those with extreme ranks, by extending known results and providing new characterizations through graph-theoretic methods.

Contribution

It proves that primitive groups synchronize maps with kernel type (k,1,...,1) and maps of ranks 3, 4, and n-2, extending previous results and offering new insights into group synchronization.

Findings

Primitive groups synchronize maps with kernel type (k,1,...,1).

Primitive groups synchronize maps of ranks 3, 4, and n-2.

Graph-theoretic techniques are used to analyze synchronization properties.

Abstract

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ synchronizes $f$ if the semigroup $\langle G,f\rangle$ contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps with kernel type $(2,1,\ldots,1)$). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is $(k,1,\ldots,1)$. Incidentally this result provides a new characterization of imprimitive groups. We also prove that the conjecture above holds for maps of extreme ranks, that is, ranks 3, 4 and $n-2$. These proofs use a graph-theoretic technique due to the second author: a transformation semigroup fails to contain a constant map if and only if it is contained in the endomorphism semigroup of a non-null (simple undircted) graph. The paper finishes with a number of open problems, whose solutions will certainly require very delicate graph theoretical considerations.