2013/06/20 by João Araüjo, Peter J. Cameron, Araújo, João +1
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1306.4827
openalex publication_date 2013/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be a set of cardinality n, G a permutation group on Ω, and f:Ω→Ω 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.