  We study the two-dimensional Wess-Zumino model with extended N=2
supersymmetry on the lattice. The lattice prescription we choose has the merit
of preserving {\it exactly} a single supersymmetric invariance at finite
lattice spacing $a$. Furthermore, we construct three other transformations of
the lattice fields under which the variation of the lattice action vanishes to
$O(ga^2)$ where $g$ is a typical interaction coupling. These four
transformations correspond to the two Majorana supercharges of the continuum
theory. We also derive lattice Ward identities corresponding to these exact and
approximate symmetries. We use dynamical fermion simulations to check the
equality of the massgaps in the boson and fermion sectors and to check the
lattice Ward identities. At least for weak coupling we see no problems
associated with a lack of reflection positivity in the lattice action and find
good agreement with theory. At strong coupling we provide evidence that
problems associated with a lack of reflection positivity are evaded for small
enough lattice spacing.
