duaLab

Bouton monoid

Contents

A Bouton monoid records enough information to calculate a chosen game value under a chosen rule for combining games.

1. Definition

Fix a monoidal operation $*$ on Games for which the underlying-position functor $U$ is lax monoidal, and a Game value $v_X:U(X)\to A$.

A Bouton monoid consists of a game value $b_X:U(X)\to B$, a monoid operation $\odot$ on $B$, and a function $a:B\to A$, such that

$$v_X=a\circ b_X,$$
$$b_{X*Y}(x,y)=b_X(x)\odot b_Y(y).$$

The pair $(x,y)$ on the left is mapped into $U(X*Y)$ by the lax monoidal comparison map. The data are minimal: from any other such data there is a surjective monoid homomorphism from a submonoid onto $B$, compatible with the game values and the maps to $A$.

See Definition 4.6.

2. Properties

Theorem 2.1 (Existence)★

For every such monoidal operation and game value, a Bouton monoid exists.

See Theorem 4.8.

3. Examples

For Conway addition and outcome, the Bouton monoid is the natural numbers with nim-sum, and the associated game value is the Grundy number. See Proposition 4.14.

4. Construction

The construction first puts a monoid structure on the positions of the terminal game, called the miniature monoid in §4.2.2. It then takes the minimum monoid factorization of the chosen value on that monoid: a monoid quotient followed by a set map, minimal among monoid factorizations.

See Definition 4.13, §4.2.3 and Appendix A.

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6. References