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 = 〈 da, db, dc 〉
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 =
〈 da + ea,
db + eb,
dc + ec 〉
d − e = 〈 da − ea, db − eb, dc − ec 〉
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 = 〈 da × k, db × k, dc × 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:
a⊐ = b
b⊐ = c
c⊐ = a
a⊏ = c
b⊏ = a
c⊏ = b
Three rotations in succession form the identity operation. Further, (p⊐)⊏ = (p⊏)⊐ = p.
d⊐ = 〈 dc, da, db 〉
The next identity, additive rotativity, is essential to the rationale for creating these algebras:
d⊐ + e⊐ = (d + e)⊐
Finally, (d + q)a = da + qa.
The remaining sections are in two branches: one for family F+ and one for family F−.