Černý's conjecture is one of the most longstanding open problems in the theory of finite automata (stated by Černý in 1964). This conjecture claims that a deterministic finite automaton with n states and synchronizing, has always a synchronizing word of length (n-1)^2. In this seminar, I am going to show an approach to Černý 's conjecture using the Wedderburn-Artin theory. First I introduce the notion of a radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. Semisimplicity gives the advantage of “factorizing” the problem of finding a synchronizing word into the sub-problems of finding words that are zeros into the projections of the simple components in the Wedderburn-Artin  decomposition. This situation is applied to prove that Černý 's conjecture holds for the class of strongly semisimple synchronizing automata: these are the automata whose sets of synchronizing words are cyclic ideals, or equivalently are ideal regular languages which are closed by takings roots.

(This is a work in progress with J. Almeida)