Congruences et Automorphismes des Automates Finis

Summary

We study a class of congruences of strongly connected finite automata, called the group congruences, which may be defined in this way: every element fixing any class of the congruence induces a permutation on this class. These congruences form an ideal of the lattice of all congruences of the automaton $\mathfrak{A}$ and we study the group associated with the maximal group congruence (maximal induced group) with respect to the Suschkevitch group of the transition monoid of $\mathfrak{A}$. The transitivity equivalence of the subgroups of the automorphism group of $\mathfrak{A}$ are found to be the group congruences associated with regular groups, which form also in ideal of the lattice of congruences of $\mathfrak{A}$. We then characterize the automorphism group of $\mathfrak{A}$ with respect to the maximal induced group. As an application, we show that, given a group G and an automaton $\mathfrak{A}$, there exists an automaton whose automorphism group is isomorphic to G and such that the quotient by the automorphism congruence is $\mathfrak{A}$.