  We consider an algebra with non-standard operations on the class of row
monomial matrices (having one unit and rest of zeros in every row). The class
of row monomial matrices is closed under multiplication, but not closed under
ordinary matrix addition. The paper considers a kind of summation operation on
row monomial matrices and the necessary conditions to be closed under the
operation in this class.
  The most significant difference between the algebra of row monomial matrices
and linear algebra is the summation operation, with respect to which the class
of row monomial matrices is closed.
  The operation of summation in the algebra can be considered also as an
algebra of subsets of any set. The class of subsets of given set is closed
under considered operation of summation.
  The deterministic finite automaton (DFA) can be presented by a complete
underlying graph of the automaton with edges labelled by letters of an
alphabet. Row monomial matrices can be induced by words in the alphabet of
labels on edges of underlying graph of DFA and present a mapping of the set of
states.
  A word $s$ of letters on edges of underlying graph $\Gamma$ of deterministic
finite automaton (DFA) is called synchronizing if $s$ sends all states of the
automaton 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, claims that $(n-1)^2$ is a precise
upper bound on the length of such a word over alphabet $\Sigma$ of letters on
edges of $\Gamma$ for every complete $n$-state DFA. The hypothesis was
formulated in 1966 by Starke.
  The following proof of the conjecture uses the algebra of row monomial
matrices.
