This work relates classes of finite automata
under various feedback products to some well-known
pseudovarieties of finite semigroups via a study of their
irreducible divisors (in the sense of Krohn-Rhodes).
In particular, this serves to relate some classical results
of Krohn, Rhodes, Stiffler, Eilenberg, Letichevsky,
Gécseg, Ésik, and Horváth.

We show that for a finite automaton satisfaction of

(1) the Letichevsky criterion for non-empty words,

(2) the semi-Letichevsky criterion for non-empty words, or

(3) neither criterion,

corresponds, respectively, to the following properties of the characteristic semigroup of the automaton:

(1) non-constructability as a divisor of
a cascade product of copies of the two-element monoid
with zero *U*,

(2) such constructability while having
*U* but no other non-trivial irreducible semigroup as
a divisor, or

(3) having no non-trivial irreducible semigroup divisors
at all.

The latter two cases are exactly the cases in
which the characteristic semigroup is *R*-trivial.

This algebraic characterization supports
the transfer of results about finite automata to
results about finite semigroups (and vice versa), and
yields insight into the lattice of pseudovarieties of
finite semigroups --- or, equivalently via the
Eilenberg correspondence, the lattice of *+*-varieties of
regular languages ---
and the operators on these
lattices that are naturally associated to
various automata products with bounded feedback.
In particular, all operators with non-trivial feedback
are shown to be equivalent, and we characterize all
pseudovarieties of finite semigroups closed under
each type of feedback product either
explicitly or by reducing the question to closure under the
cascade product.

Sponsored in part by the FCT approved
projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with
the European Community Fund FEDER and by FCT through *Centro de Matemática
da Universidade do Porto.* Also sponsored in part by FCT, the
*Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III*, and by *Caixa Geral de Depósitos*.