On the Lattice Structure of Finite Games
Dan Calistrate, Marc M. Paulhus, David Wolfe · 2002
. We prove that games born by day n form a distributive lattice, but that the collection of all finite games does not form a lattice. Introduction A great deal is known about the partial order structure of large subsets of games. See, for instance, [BCG82] [Con76] for a complete characterization of games generated by numbers, and infinitesimals such as and *n. Linear operators applied to these games of temperature zero can often leverage this characterization to apply to hot games, such as positions occurring in Go [BW94] and Domineering [Ber88] [Wol93]. Some general results are known about the group structure of games, including a complete characterization of the group generated by games born by day 3 [Moe91], but surprisingly little has been written about the overall structure of the partial-ordering of games. Here we prove that the games born by day n form a distributive lattice, but that the collection of all finite games do not form a lattice. We assume the reader is already familiar with combinatorial game theory definitions as in [BCG82] or [Con76]. In particular, we assume knowledge of the definitions of a game [BCG82, p. 21], sums and negatives of games [BCG82, p. 33], and the standard partial ordering on games [BCG82, p. 34].