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−.