On Direct Product and Quotient of Strongly Connected Automata

Let $A \ \times \ B$ be the direct product of a strongly connected permutation automaton $A$ and a strongly connected synchronizing (reset) automaton $B$, then $A \ \times \ B$ is strongly connected and $$\boldsymbol{A \cong (A \times B)/\pi}$$ $$\boldsymbol{B \cong (A \times B)/\rho}$$ $$\boldsymbol{(A \times B) \ \cong \ (A \times B)/\pi \ \times \ (A \times B)/\rho}$$ where $\pi$ and $\rho$ are automaton congruence relations defined in this paper, $(A \times B)/\pi$ and $(A \times B)/\rho$ are quotient automata constructed by $\pi$ and $\rho$ respectively.