  It is known that graded cyclic modules over $S=K[x,y]$ have the Weak
Lefschetz Property (WLP). This is not true for non-cyclic modules over $S$. The
purpose of this note is to study which conditions on $S$-modules ensure the
WLP. We give an algorithm to test the WLP for graded modules with fixed Hilbert
function. In particular, we prove that indecomposable graded modules over $S$
with the Hilbert function $(h_0,h_1)$ have the WLP.
