Nimber

In mathematics, the proper class of nimbers (occasionally called Grundy numbers) is introduced in combinatorial game theory, where they are defined as the values of nim heaps, but arise in a much larger class of games because of the Sprague–Grundy theorem. It is the proper class of ordinals endowed with a new nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.

Properties

The Sprague–Grundy theorem states that every impartial game is equivalent to a nim heap of a certain size. Nimber addition (also known as nim-addition) can be used to calculate the size of a single heap equivalent to a collection of heaps. It is defined recursively by

$\alpha + \beta = \operatorname{mex}(\{\,\alpha' + \beta : \alpha' < \alpha\,\} \cup \{\, \alpha + \beta' : \beta' < \beta \,\}),$

where for a set S of ordinals, mex(S) is defined to be the "minimum excluded ordinal", i.e. mex(S) is the smallest ordinal which is not an element of S. For finite ordinals, the nim-sum is easily evaluated on computer by taking the exclusive-or of the corresponding numbers (whereby the numbers are given their binary expansions, and the binary expansion of (x xor y) is evaluated bit-wise).

Nimber multiplication (nim-multiplication) is defined recursively by

α β = mex{α ′ β + α β ′ − α ′ β ′ : α ′ < α, β ′ < β} = mex{α ′ β + α β ′ + α ′ β ′ : α ′ < α, β ′ < β}.

Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal α is α itself. The nimber multiplicative inverse of the nonzero ordinal α is given by 1/α = mex(S), where S is the smallest set of ordinals (nimbers) such that

1. 0 is an element of S;
2. if 0 < α ′ < α and β ′ is an element of S, then [1 + (α ′ − α) β ′ ]/α ′ is also an element of S.

For all natural numbers n, the set of nimbers less than 2²ⁿ form the Galois field GF(2²ⁿ) of order 2²ⁿ.

In particular, this implies that the set of finite nimbers is isomorphic to the direct limit of the fields GF(2²ⁿ), for each positive n. This subfield is not algebraically closed, however.

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that

1. The nimber product of distinct Fermat 2-powers (numbers of the form 2²ⁿ) is equal to their ordinary product;
2. The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers.

The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal ω^ωω, where ω is the smallest infinite ordinal. It follows that as a nimber, ω^ωω is transcendental over the field.

Addition and multiplication tables

The following tables exhibit addition and multiplication among the first 16 nimbers.
This subset is closed under both operations, since 16 is of the form 2²ⁿ (When you prefer simple text tables - they are here.)

Nimber addition (sequence A003987 in OEIS)
This is also the Cayley table of Z₂⁴ - or the table of bitwise XOR operations.
The small matrices show the single digits of the binary numbers.

Nimber multiplication (sequence A051775 in OEIS)
The nonzero elements form the Cayley table of Z₁₅ (as this different arrangement spells out).
The small matrices differ only by exchanged rows from this one, showing XOR operations.

References

• Conway, John Horton (1976). On Numbers and Games. Academic Press Inc. (London) Ltd.. 
• Lenstra, H. W. (1978). "Nim multiplication". hdl:1887/2125. 
• Schleicher, Dierk; Stoll, Michael. "An Introduction to Conway's Games and Numbers". arXiv:math.DO/0410026.  which discusses games, surreal numbers, and nimbers.