Nim Addition and Split Extensions basis for By

Nim Addition and Split Extensions basis for By Lydia Njuguna and Benard Kivunge Kenyatta University, Kenya Third Mile High Conference on Nonassociative Mathematics 13 th August 2013

Introduction A sequence of algebras over the field of real numbers can be constructed, each with twice the dimension of the previous one The oldest method of constructing these algebras is the Cayley. Dickson formula and the algebras thus produced are known as Cayley. Dickson algebras

• The algebra constructed by doubling complex numbers is the quaternions • Next we have the octonions, constructed by forming ordered pairs of quaternions • The algebra immediately following the octonions is the sedenions

Real ons Real Complex ons Complex Quaternion Octonion Sedenion ons

Multiplication of Spit Extensions Let L be a multiplicative subloop of the non-zero octonions. Then its sedenion extension is the disjoint union within the sedenions. Elements of this union are encoded as pairs with , by The multiplication of elements of in general is given by the following equations:

Nim Addition Consider the non-negative integers {0, 1, 2, 3, …}. Nim addition and multiplication gives a way of defining addition and multiplication in to make it a field of characteristic 2. The rules of Nim addition simplify to the form: The Nim-sum of a number of distinct powers of 2 is the ordinary sum. The Nim-sum of two equal numbers is 0 Example: 32 +16 + 4 + 1 = 53

Complex split extensions

Table 1 Multiplication of Complex Split Extensions

The following table gives the Nim addition for the elements from 0 to 3 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 Table 2 Nim addition for the elements 0 to 3

This table is similar to the multiplication table of the complex split extensions (Table 1). The subscripts of the elements in Table 1 make up Table 2. Observation: k, m = 0, 1, 2, 3

Quaternion Split Extensions

Table 3 Mult of Quaternion Split Extention

The following table gives the Nim addition for the elements from 0 to 7

0 1 2 3 4 5 6 7 0 0 1 2 3 4 5 6 7 1 1 0 3 2 5 4 7 6 2 2 3 0 1 6 7 4 5 3 3 2 1 0 7 6 5 4 4 4 5 6 7 0 1 2 3 5 5 4 7 6 1 0 3 2 6 6 7 4 5 2 3 0 1 7 7 6 5 4 3 2 1 0 Table 4 Nim addition for 0 to 7

This table is similar to the multiplication table of the quaternion split extensions (Table 3). The subscripts of the elements in Table 3 make up Table 4. Observation k, m = 0, 1, . . . , 7

Octonion Split Extensions be basis elements in The multiplication of the elements give rise to the following table (only subscripts shown)

The following table gives the Nim addition for the elements from 0 to 15

The table is similar to the multiplication table of the octonion split extensions 5. The subscripts of the elements in Table 5 make up Table 6

Observation From tables 5 and 6, the following observations can be made: There are 4 cases

Case 2:

3. 4 Sedenion Split Extensions

Using the three results of the previous Section the multiplication of these elements can be summarized in the following 4 cases Case 1:

Case 3:

Case 4:

In general,

4. Main Theorem Consider the split extension basis elements of dimension , n = 1, 2, …given by

Proof:

There are 4 cases Case 1:

Case 2:

Case 3:

Case 4:

The end Thank you