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A word w of letters on edges of underlying graph Gamma of deterministicfinite automaton (DFA) is called the synchronizing word if w sends all statesof the automaton to a unique state. J. Cerny discovered in 1964 a sequence ofn-state complete DFA possessing a minimal synchronizing word of length (n-1)^2.The hypothesis, well known today as the Cerny conjecture, claims that it isalso precise upper bound on the length of such a word for a complete DFA. Thissimple looking conjecture is arguably the most fascinating and longstandingopen problem in the theory of finite automaton. The hypothesis was formulatedin 1966 by Starke. The problem has motivated great and constantly growingnumber of investigations and generalizations. To prove the conjecture, we usealgebra with non-standard operations on a special class of matrices (rowmonomial), induced by words in the alphabet of labels on edges. These matricesgenerate a space with respect to the mentioned operation. The proof is based onconnection between length of words u and dimension of the space generated bysolutions Lx of matrix equation MuLx =Ms for synchronizing word s, as well ason the relation between ranks of Mu and Lx.

The length of a minimal synchronizing word and the \v{C}erny conjecture