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 ϕ : X → Y such that x ≤ ϕ(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–Holder 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∣.
['Gábor Czédli', 'Benoit Larose', 'György Pollák']
Notes on Coalition Lattices