21 research outputs found
The averaging trick and the Cerny conjecture
The results of several papers concerning the \v{C}ern\'y conjecture are
deduced as consequences of a simple idea that I call the averaging trick. This
idea is implicitly used in the literature, but no attempt was made to formalize
the proof scheme axiomatically. Instead, authors axiomatized classes of
automata to which it applies
A quadratic upper bound on the size of a synchronizing word in one-cluster automata
International audienceČerný's conjecture asserts the existence of a synchronizing word of length at most (n-1)² for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p, q, one has p*ar = q*as for some integers r, s (for a state p and a word w, we denote by p*w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n²). This applies in particular to Huffman codes
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
Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks
Let be a set of cardinality , a permutation group on
, and a map which is not a permutation. We say that
synchronizes if the semigroup 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 primitive groups synchronize maps of rank (thus, maps
with kernel type ). We prove some extensions of Rystsov's
result, including this: a primitive group synchronizes every map whose kernel
type is . 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 .
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