  We introduce filtered cohomologies of differential forms on symplectic
manifolds. They generalize and include the cohomologies discussed in Paper I
and II as a subset. The filtered cohomologies are finite-dimensional and can be
associated with differential elliptic complexes. Algebraically, we show that
the filtered cohomologies give a two-sided resolution of Lefschetz maps, and
thereby, they are directly related to the kernels and cokernels of the
Lefschetz maps. We also introduce a novel, non-associative product operation on
differential forms for symplectic manifolds. This product generates an
A-infinity algebra structure on forms that underlies the filtered cohomologies
and gives them a ring structure. As an application, we demonstrate how the ring
structure of the filtered cohomologies can distinguish different symplectic
four-manifolds in the context of a circle times a fibered three-manifold.
