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 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. In particular, the set of states of the automaton might be considered. 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. The algebra under consideration plays an important role in the study of DFA, especially for synchronizing automata.