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 unit order. 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 bidual of $\cal X$. These results are then applied to several extensions of the classical optimal transport. In particular, an alternate proof of Kellerer's result is given without using the Choquet Theorem.
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