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When do coalitions form a lattice? (English) Zbl 0829.06003

Summary: Given a finite partially ordered set \(P\), for subsets or, in other words, coalitions \(X\), \(Y\) of \(P\) let \(X\leq Y\) mean that there exists an injection \(\varphi: X\to Y\) such that \(x\leq \varphi (x)\) for all \(x\in X\). The set \({\mathcal L} (P)\) of all subsets of \(P\) equipped with this relation is a partially ordered set. All partially ordered sets \(P\) such that \({\mathcal L} (P)\) is a lattice are determined, and this result is extended to quasiordered sets \(P\) versus \(q\)-lattices \({\mathcal L} (P)\) as well. Some elementary properties of distributive lattices \({\mathcal L} (P)\) are also given.

MSC:

06A06 Partial orders, general
06B99 Lattices
91A99 Game theory
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