Section 1. This report introduces two families of algebras over the ordered triples of integers. The names of the families are F+ and F− for reasons that will become clear.
The triples are written within shallow angle brackets, as ⟨ +4, +2, −3 ⟩. Three constant triples, termed atoms, get special names:
a = ⟨ 1, 0, 0 ⟩
b = ⟨ 0, 1, 0 ⟩
c = ⟨ 0, 0, 1 ⟩
The integer components of a triple can be selected by using subscripts a, b, and c. Meanwhile, variables typically have names such as d, e, and f. Here is a key notational identity:
d = ⟨ d_{a}, d_{b}, d_{c} ⟩
It is often helpful to think of a triple as a linear combination of constants a, b, and c.
Section 2. In both familes, addition and subtraction are in parallel:
d + e =
⟨ d_{a} + e_{a},
d_{b} + e_{b},
d_{c} + e_{c} ⟩
d − e =
⟨ d_{a} − e_{a},
d_{b} − e_{b},
d_{c} − e_{c} ⟩
Addition is commutative and associative:
d + e = e + d
(d + e) + f = d + (e + f)
The additive identity is 0 = ⟨ 0, 0, 0 ⟩.
Multiplication by an integer k is routine:
d × k = k × d = ⟨ d_{a} × k, d_{b} × k, d_{c} × k ⟩
The additive inverse is −d = −1 × d.
Addition therefore forms an abelian group.
Section 3. There is a rotation operation and its inverse for the triples. The operations are symbolized by a superscript square missing one side, and are defined thus:
rotation
a^{⊐} = b b^{⊐} = c c^{⊐} = a | inverse rotation a^{⊏} = c b^{⊏} = a c^{⊏} = b |
Three rotations in succession form the identity operation. Further, (p^{⊐})^{⊏} = (p^{⊏})^{⊐} = p.
This follows:
d^{⊐} = ⟨ d_{c}, d_{a}, d_{b} ⟩
The next identity, additive rotativity, is essential to the rationale for creating these algebras:
d^{⊐} + e^{⊐} = (d + e)^{⊐}
Finally, (d + q)_{a} = d_{a} + q_{a}.
The remaining sections are in two branches: one for family F+ and one for family F−.