Two families of algebras over ordered triples of integers.
Version of Monday 29 July 2019.
Dave Barber's other pages.

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
de = ⟨ daea, dbeb, dcec

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:

 rotation a⊐ = b b⊐ = c c⊐ = a inverserotation a⊏ = c b⊏ = a c⊏ = b

Three rotations in succession form the identity operation. Further, (p) = (p) = p.

This follows:

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−.