When Do Coalitions Form a Lattice?

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. All partially ordered sets P such that L(P ) is a lattice are determined, and this result is extended to quasiordered set P versus q-lattice L(P ) as well. Some elementary properties of distributive lattices L(P ) are also given. Dedicated to Professors László Leindler on his 60th and Károly Tandori on his 70th birthday Motivation and preliminaries In game theory or in the mathematics of human decision making the following situation is frequently considered, cf. e.g. Peleg [5]. Given a finite set P , for example we may think of P as a set of political parties, and each x ∈ P has a certain strength measured on a numerical scale that we may think of as the number of votes x receives. Subsets of P are called coalitions. The strength of a coalition is the sum of strengths of its members. Let L(P ) stand for the set of all coalitions. The relation “stronger or equally strong” is a quasiorder on P and also on L(P ). The quasiorder on P has some influence on the quasiorder on L(P ). Sometimes, like before the election in our example, all we have is a quasiorder or, more frequently, a partial order on P , supplied e.g. by a public opinion poll. Yet, as we will see, this often suffices to build some algebraic structure on L(P ). From now on, let P = 〈P,≤〉 be a fixed finite quasiordered set, i.e., ≤ is a reflexive and transitive relation on the finite set P . For x, y ∈ P , x > y means that y ≤ x and x 6≤ y. For undefined terminology the reader is referred to Grätzer [4]. Even without explicit mentioning, all sets occurring in this paper are assumed to be finite. The set of all subsets, alias coalitions, of P is denoted by L(P ). For X, Y ∈ L(P ), a map φ: X → Y is called an extensive map if φ is injective and for every x ∈ X we have x ≤ φ(x). Let X ≤ Y mean that there exists an extensive map X → Y ; this definition turns L(P ) into a quasiordered set L(P ) = 〈L(P ),≤〉. Using singleton coalitions one can easily see that P is a partially ordered set, in short a poset, iff L(P ) is a poset. Our main result, Thm. 2, describes the posets P 1991 Mathematics Subject Classification. Primary 06B99, Secondary 06A99, 90D99.