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.
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