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A word $s$ of letters on edges of underlying graph $\Gamma$ of deterministicfinite automaton (DFA) is called synchronizing if $s$ sends all states of theautomaton to a unique state. J. \v{C}erny discovered in 1964 a sequence of$n$-state complete DFA possessing a minimal synchronizing word of length$(n-1)^2$. The hypothesis, mostly known today as \v{C}erny conjecture, claimsthat $(n-1)^2$ is a precise upper bound on the length of such a word overalphabet $\Sigma$ of letters on edges of $\Gamma$ for every complete $n$-stateDFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandardoperation over special class of matrices induced by words in the alphabet oflabels on edges is used to prove the conjecture. The proof is based on theconnection between length of words $u$ and dimension of the space generated bysolution $L_x$ of matrix equation $M_uL_x=M_s$ for synchronizing word $s$, aswell as on relation between ranks of $M_u$ and $L_x$. Important role belowplaced the notion of pseudo inverseL matrix, sometimes reversible.

Cerny-Starke conjecture from the sixties of XX century