  The classical duality theory of Kantorovich and Kellerer for the classical
optimal transport is generalized to an abstract framework and a
characterization of the dual elements is provided. This abstract generalization
is set in a Banach lattice $\cal{X}$ with a order unit. The primal problem is
given as the supremum over a convex subset of the positive unit sphere of the
topological dual of $\cal{X}$ and the dual problem is defined on the bi-dual of
$\cal{X}$. These results are then applied to several extensions of the
classical optimal transport.
