Partitions versus sets: A case of duality

In a recent paper, Amini et al. introduced a general framework to prove duality theorems between tree-decompositions and their dual combinatorial object. They unify all known ad hoc proofs in one duality theorem based on submodular partition functions. This general theorem remains however a bit technical and relies on this particular submodularity property. Instead of partition functions, we propose here a simple combinatorial property of a set of partitions which also gives these duality results. Our approach is both simpler, and a little bit more general.