Structures of sublattices related to Veinott relation

Let E be a nonempty finite set. Narayanan showed a theorem describing that the family {Π′|Π′ ∈ PE, ΣX∈Π′ f(X) = minΠ∈PEΣX∈Π f(X)} forms a lattice, where f is a submodular function on 2^E and PE is the set of all partitions of E. On the other hand, Shapley gave a theorem on a necessary and sufficient condition for a convex game to be decomposable. We give a theorem which is a generalization of these two theorems.