  We define tropical analogues of the notions of linear space and Plucker
coordinate and study their combinatorics. We introduce tropical analogues of
intersection and dualization and define a tropical linear space built by
repeated dualization and transverse intersection to be constructible. Our main
result that all constructible tropical linear spaces have the same f-vector and
are ``series-parallel''. We conjecture that this f-vector is maximal for all
tropical linear spaces with equality precisely for the series-parallel tropical
linear spaces. We present many partial results towards this conjecture.
  In addition we relate tropical linear spaces to linear spaces defined over
power series fields and give many examples and counter-examples illustrating
aspects of this relationship. We describe a family of particularly nice
series-parallel linear spaces, which we term tree spaces, that realize the
conjectured maximal f-vector and are constructed in a manner similar to the
cyclic polytopes.
