Groups Synchronizing a Transformation of Non-Uniform Kernel

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 $$ 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

Primitive groups, graph endomorphisms and synchronization

The third author has been partially supported by the Fundação para a Ciência e a Tecnologia through the project CEMAT-CIÊNCIAS UID/Multi/04621/2013.Let Ω be a set of cardinality n, G be a permutation group on Ω and f:Ω→Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup ⟨G,f⟩ contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation.  A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.  The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n−1 and n−2, and here this is extended to n−3 and n−4.  In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically √|A|. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.PostprintPeer reviewe

Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks

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.Comment: Includes changes suggested by the referee of the Journal of Combinatorial Theory, Series B - Elsevier. We are very grateful to the referee for the detailed, helpful and careful repor

Permutation groups and transformation semigroups : results and problems

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S= such as regularity and generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.PostprintPeer reviewe

Between primitive and 2-transitive : synchronization and its friends

The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe