Notes on Coalition Lattices |
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Authors: | Gábor Czédli Benoit Larose György Pollák |
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Institution: | (1) JATE Bolyai Institute, Szeged, Aradi vértanúk tere 1, H–6720, Hungary;(2) Champlain Regional College, 900 Riverside Drive, St-Lambert, Qc, J4P 3P2, Canada;(3) Mathematical Research Institute, Budapest, Hungary |
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Abstract: | Given a finite partially ordered set P, for subsets or, in other words coalitions X, Y of P let X Y mean that there exists an injection : X Y such that x (x) for all x X. The set L(P) of all subsets of P equipped with this relation is a partially ordered set. When L(P) is a lattice, it is called the coalition lattice of P. It is shown that P is determined by the coalition lattice L(P). Further, any coalition lattice satisfies the Jordan–Hölder chain condition. The so-called winning coalitions, i.e. coalitions X such that P\X X in L(P), are shown to form a dual ideal in L(P). Finally, an inductive formula on P is given to describe the lattice operations in L(P), and this result also works for certain quasiordered sets P . |
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Keywords: | coalition coalition lattice lattice partially ordered set quasiorder winning coalition |
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