This book develops a theory of formal power series in noncommuting
variables, the main emphasis being on results applicable to automata and
formal language theory. This theory was initiated around 196O-apart from
some scattered work done earlier in connection with free groups-by M. P.
Schutzenberger to whom also belong some of the main results. So far
there is no book in existence concerning this theory. This lack has had
the unfortunate effect that formal power series have not been known and
used by theoretical computer scientists to the extent they in our
estimation should have been. As with most mathematical formalisms, the
formalism of power series is capable of unifying and generalizing known
results. However, it is also capable of establishing specific results
which are difficult if not impossible to establish by other means. This
is a point we hope to be able to make in this book. That formal power
series constitute a powerful tool in automata and language theory
depends on the fact that they in a sense lead to the arithmetization of
automata and language theory. We invite the reader to prove, for
instance, Theorem IV. 5. 3 or Corollaries III. 7. 8 and III. 7.- all
specific results in language theory-by some other means. Although this
book is mostly self-contained, the reader is assumed to have some
background in algebra and analysis, as well as in automata and formal
language theory.