ABSTRACT: Under Conway's simple but powerful game theory axioms, games form a group with a partial order. While a great deal has been known about the group structure of large subsets of games, surprisingly little was known about the overall partial order. We prove that games lasting a fixed number of turns form a distributive lattice, but that the collection of all finite games does not form a lattice.

We will also present theorems about the structure of this lattice. A direct corollary of these theorems is that all maximal chains in the day n lattice are of the same length, that length being exactly one plus twice the number of games born by day n-1.

This talk will be introductory and I'll explain the relevant game theory and lattice theory as we go.