  This paper can be seen as a continuation of the works contained in the recent
preprints [Za], of the second author, and [Mi], of Juan Migliore. Our results
are:
  1). There exist codimension three artinian level algebras of type two which
do not enjoy the Weak Lefschetz Property (WLP). In fact, for $e\gg 0$, we will
construct a codimension three, type two $h$-vector of socle degree $e$ such
that {\em all} the level algebras with that $h$-vector do not have the WLP. We
will also describe the family of those algebras and compute its dimension, for
each $e\gg 0$.
  2). There exist reduced level sets of points in ${\mathbf P}^3$ of type two
whose artinian reductions all fail to have the WLP. Indeed, the examples
constructed here have the same $h$-vectors we mentioned in 1).
  3). For any integer $r\geq 3$, there exist non-unimodal monomial artinian
level algebras of codimension $r$. As an immediate consequence of this result,
we obtain another proof of the fact (first shown by Migliore in [Mi], Theorem
4.3) that, for any $r\geq 3$, there exist reduced level sets of points in
${\mathbf P}^r$ whose artinian reductions are non-unimodal.
