Automorphism groups and quotients of strongly connected automata and monadic algebras

The class of total automata is characterized. Relationships between the structure of the automorphism group G(A) of a finite automaton A and G(A/H), where A/H is a quotient [9] of A, are exhibited. It is shown that the poset PA of isomorphism classes of quotients of A is an antihomomorphic, image of the poset PG(A) of conjugacy classes of subgroups of G(A). Some results are obtained about natural series of quotient automata. Applications to decomposition theory, in particular to the problem of factoring out identical parallel front components, are given. A generalization of the major parts of the theory to infinite strongly connected monadic algebras is obtained.