  We extend the Larson-Sweedler theorem to weak Hopf algebras by proving that a
finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a
non-degenerate left integral. We show that the category of modules over a weak
Hopf algebra is autonomous monoidal with semisimple unit and invertible
modules. We also reveal the connection of invertible modules to left and right
grouplike elements in the dual weak Hopf algebra. Defining distinguished left
and right grouplike elements we derive the Radford formula for the fourth power
of the antipode in a weak Hopf algebra and prove that the order of the antipode
is finite up to an inner automorphism by a grouplike element in the trivial
subalgebra A^T of the underlying weak Hopf algebra A.
