  In this thesis, we study the place of regular languages within the
communication complexity setting. In particular, we are interested in the
non-deterministic communication complexity of regular languages.
  We show that a regular language has either O(1) or Omega(log n)
non-deterministic complexity. We obtain several linear lower bound results
which cover a wide range of regular languages having linear non-deterministic
complexity. These lower bound results also imply a result in semigroup theory:
we obtain sufficient conditions for not being in the positive variety Pol(Com).
  To obtain our results, we use algebraic techniques. In the study of regular
languages, the algebraic point of view pioneered by Eilenberg (\cite{Eil74})
has led to many interesting results. Viewing a semigroup as a computational
device that recognizes languages has proven to be prolific from both semigroup
theory and formal languages perspectives. In this thesis, we provide further
instances of such mutualism.
