Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks
Araújo, João; Cameron, Peter
Journal of Combinatorial Theorie, Series B, 106 (2014), 98-114
http://dx.doi.org/10.1016/j.jctb.2014.01.006 (preprint - http://arxiv.org/abs/1306.4827)
Let ? be a set of cardinality n, G a permutation group on ?, and f:??? a map which is not a permutation. We say that G synchronizes f if the semigroup ?G,f? 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,…,1)). We prove some extensions of Rystsov's result, including this: a primitive group synchronizes every map whose kernel type is (k,1,…,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.