The Abstract Structure of the Group of Games

Summary

ABSTRACT. We compute the abstract group structure of the group Ug of partizan games and the group ShUg of short partizan games. We also determine which partially ordered cyclic groups are subgroups of Ug and ShUg.

As in [2], let Ug be the group of all partizan combinatorial games, let No be the field of surreal numbers, and for G in Ug, let L(G) and R(G) be the Left and Right sections of G, respectively. If L(G) is the section just to the left or right of some number z, we say that z is the Left stop of G, and similarly for R(G) and the Right stop. Let ShUg be the group of all short games in Ug; that is, ShUg is the set of all games born before day a;, or of all games which can be expressed in a form with only finitely many positions. For games U and integers n, we write

Also, recall from [1, Chapter 8] the definition of Norton multiplication, for a game G and a game U > 0:

Here, G^L, G^R, U^L, and U^R range independently over the left options of G, right options of G, left options of U, and right options of U, respectively. To define G.U, we must fix a form of G and sets of Left and Right options for U.

We will say that a subgroup X of Ug has the integer translation property if it contains the integers and, whenever either Left or Right has a winning move in a sum A₁+…+A_n of games from X, not all integers, he also has a winning move in an A_j which is not equal to an integer.

Lemma 1. The real numbers have the integer translation property.