  In two articles by Barthel, Brasselet, Fieseler and Kaup, and, Bressler and
Lunts, a combinatorial theory of intersection cohomology and perverse sheaves
has been developed on fans. In the first one, one tried to present everything
on an elementary level,using only some commutative algebra and no derived
categories. There remained two major gaps:
  First of all the Hard Lefschetz Theorem was only conjectured and secondly the
intersection product seemed to depend on some non-canonical choices. Meanwhile
the Hard Lefschetz theorem has been proved by Karu. The proof relies heavily on
the intersection product, since what finally has to be shown are the
Hodge-Riemann relations. In fact here again choices enter: The intersection
product is induced from the intersection product on some simplicial subdivision
via a direct embedding of the corresponding intersection cohomology sheaves, a
fact, which makes the argumentation quite involved. In a recent paper by
Bressler and Lunts, one shows by a detailed analysis that eventually all
possible choices do not affect the definition of the pairing.
  Our goal here is the same, but we shall try to follow the spirit of the first
cited paper avoiding the formalism of derived categories. For perverse sheaves
we define their dual sheaf and check that the intersection cohomology sheaf is
self- dual in a natural way.
