ON THE CHARACTERISTIC S-AUTOMATA

In this paper we shall discuss some properties derived from the characteristic S-automaton $_x(S)_M$, using the fact that ${\mu}_S$ is an equivalence relation on M. When $L_{m}:S{\rightarrow}M$ is a left translation and $L_{M}$ is a collection of $L_{m}'s$, we shall show $_x(S)_{M}{\cong}L_{M}$. If S is commutative, we have $_x(S)_{M{\times}N{\cong}L_{M{\times}N}$. Moreover when M and N are perfect, we have $L_{M{\times}N}{\cong}L_{M}{\times}L_{N}$ and $_x(S)_{M{\times}N}{\cong}_x(S)_{M}{\times}_x(S)_N$.