Coalition lattices
by
Gábor Czédli
University of Szeged

Given a partially ordered set P, always finite, the set of all subsets, alias coalitions , of P will be denoted by L(P). For X, Y in L(P), a map \phi: X --> Y is called an extensive map if \phi is injective and for every x in X we have x <= \phi(x). Let X <= Y mean that there exists an extensive map X --> Y; this definition turns L(P) into a partially ordered set. It is a lattice, a so-called coalition lattice , iff P is a forest, i.e., no two incomparable elements of P has an upper bound. There are three ways of describing lattice operations (given by Czédli, Pollák, Davidson and Grätzer) but none of them is so simple as in related lattices usually. The lattice L(P) satisfies the Jordan-Hölder chain condition, and it determines P up to isomorphism. L(P) is distributive iff modular iff all the trees of the forest P are chains. Each coalition lattice L(P) has a winning coalition , i.e. an X in L(P) with X <= P\X. It is an open problem if the class of all coalition lattices satisfies a nontrivial lattice identity. All we know that it satisfies nontrivial Horn sentences, e.g. the Horn sentence which expresses that M₃ is not a sublattice.

References

G. Czédli and Gy. Pollák, When do coalitions form a lattice?, Acta Sci. Math. (Szeged), 60 (1995), 197-206.

G. Czédli, A Horn sentence in coalition lattices, Acta Math. Hungarica 72 (1996), 99-104.

G. Czédli, B. Larose and Gy. Pollák, Notes on coalition lattices, Order 16 (1999) 19-29.

M. Davidson and G. Grätzer, A note on coalitions, Acta Sci. Math. (Szeged), 61, 33-34.

Date received: May 21, 2004