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Structures of sublattices related to Veinott relation. (English) Zbl 0883.90134

Summary: Let \(E\) be a nonempty finite set. H. Narayanan showed a theorem describing that the family \(\left\{\Pi'\mid\Pi'\in P_E, \sum_{X\in\Pi'} f(X)= \min_{\Pi\in P_E} \sum_{X\in\Pi'} f(X)\right\}\) forms a lattice, where \(f\) is a submodular function on \(2^E\) and \(P_E\) is the set of all partitions of \(E\). On the other hand, L. S. 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.

MSC:

91A12 Cooperative games
06B99 Lattices
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