Discussion of family F−.

Section 4−. Each of the algebras within family F− has a different definition of multiplication, but within each the following properties must always be satisfied.

Multiplication is anti-commutative:

d × e = − e × d

Multiplication is not associative, but it does satisfy the flexible identity:

a × (b × a) = (a × b) × a

Multiplication also satisfies the Jabobi identity:

d × (e × f) + e × (f × d) + f × (d × e) = 0

Multiplication distributes over addition on both sides:

(d + e) × f = d × f + e × f
f × (d + e) = f × d + f × e

Zero is absorbent: 0 × d = 0.

Because of anti-commutativity, there cannot be an identity for multiplication.

Multiplicative rotativity is like that of the F+ family:

d × e = (d × e)

The product of two atoms is not necessarily an atom, but will be a linear combination of atoms.

With addition and multiplication as defined here, each algebra of family F− is a Lie algebra.

Finally, (d × q)a is not required to equal da × qa.

Section 5−. Because multiplication distributes over addition, multiplication in all the algebras in F− can be defined by these nine atomic products:

 a × a b × a c × a a × b b × b c × b a × c b × c c × c

By anti-commutativity, a × a = b × b = c × c = 0.

Let t, u, v be integer constants. If a × b = ⟨ t, u, v ⟩ then by rotativity b × c = ⟨ v, t, u ⟩ and c × a = ⟨ u, v, t ⟩.

By anti-commutativity, b × a = − ⟨ t, u, v ⟩, c × b = − ⟨ v, t, u ⟩ and a × c = − ⟨ u, v, t ⟩.

To summarize:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = − ⟨ t, u, v ⟩ c × a = + ⟨ u, v, t ⟩ a × b = + ⟨ t, u, v ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = − ⟨ v, t, u ⟩ a × c = − ⟨ u, v, t ⟩ b × c = + ⟨ v, t, u ⟩ c × c = ⟨ 0, 0, 0 ⟩

Incidentally,

(a × c) = − a × b
(b × a) = − b × c
(c × b) = − c × a

A script in the F− case can reduce to an ordered quadruple, with a semicolon after the initial zero:

⟨ 0; t, u, v

Equivalently in full:

⟨ 0; (a × b)a, (a × b)b, (a × b)c

Equivalently in brief:

⟨ 0; a × b

Section 6−. The simplest non-trivial multiplications in family F− are the one shown immediately below, and its negative. The one shown is named SM− (for simple multiplication), and its script is ⟨ 0; 0, 0, +1 ⟩:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ 0, 0, −1 ⟩ c × a = ⟨ 0, +1, 0 ⟩ a × b = ⟨ 0, 0, +1 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ −1, 0, 0 ⟩ a × c = ⟨ 0, −1, 0 ⟩ b × c = ⟨ +1, 0, 0 ⟩ c × c = ⟨ 0, 0, 0 ⟩

More compactly:

a × a = b × b = c × c = 0
a × b = c = −b × a
b × c = a = −c × b
c × a = b = −a × c

The script of SM−'s negative simply inverts all the signs: ⟨ 0; 0, 0, +1 ⟩.

Applying a constant factor to all the products yields a valid but different multiplication — and consequently a different algebra. The same effect can also be achieved by applying the constant factor to each term of the script. The following example is the quintuple of SM−:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ 0, 0, −5 ⟩ c × a = ⟨ 0, +5, 0 ⟩ a × b = ⟨ 0, 0, +5 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ −5, 0, 0 ⟩ a × c = ⟨ 0, −5, 0 ⟩ b × c = ⟨ +5, 0, 0 ⟩ c × c = ⟨ 0, 0, 0 ⟩

If that constant factor happens to be zero, the result becomes trivial. Two trivial F− multiplications.

Here are four examples of non-trivial multiplications:

Script ⟨ 0; +1, 0, −1 ⟩:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ −1, 0, +1 ⟩ c × a = ⟨ 0, −1, +1 ⟩ a × b = ⟨ +1, 0, −1 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ +1, −1, 0 ⟩ a × c = ⟨ 0, +1, −1 ⟩ b × c = ⟨ −1, +1, 0 ⟩ c × c = ⟨ 0, 0, 0 ⟩

Script ⟨ 0; +1, −6, +5 ⟩:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ −1, +6, −5 ⟩ c × a = ⟨ −6, +5, +1 ⟩ a × b = ⟨ +1, −6, +5 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ −5, −1, +6 ⟩ a × c = ⟨ +6, −5, −1 ⟩ b × c = ⟨ +5, +1, −6 ⟩ c × c = ⟨ 0, 0, 0 ⟩

Script ⟨ 0; +2, −5, +3 ⟩:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ −2, +5, −3 ⟩ c × a = ⟨ −5, +3, +2 ⟩ a × b = ⟨ +2, −5, +3 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ −3, −2, +5 ⟩ a × c = ⟨ +5, −3, −2 ⟩ b × c = ⟨ +3, +2, −5 ⟩ c × c = ⟨ 0, 0, 0 ⟩

Script ⟨ 0; +5, +2, −7 ⟩:

 a × a = ⟨ 0, 0, 0 ⟩ b × a = ⟨ −5, −2, +7 ⟩ c × a = ⟨ +2, −7, +5 ⟩ a × b = ⟨ +5, +2, −7 ⟩ b × b = ⟨ 0, 0, 0 ⟩ c × b = ⟨ +7, −5, −2 ⟩ a × c = ⟨ −2, +7, −5 ⟩ b × c = ⟨ −7, +5, +2 ⟩ c × c = ⟨ 0, 0, 0 ⟩

Section 7−. Provided as an extended set of examples, the scripts below represent all the valid multiplications within family F− meeting the following four criteria:

• No number has an absolute value greater than five.
• The numbers have no common factor greater than one.
• The first number after the semicolon is not negative.

 ⟨ 0; 0, −1, +1⟩ ⟨ 0; 0, 0, −1⟩ ⟨ 0; 0, 0, +1⟩ ⟨ 0; 0, +1, −1⟩ ⟨ 0; +1, −5, +4⟩ ⟨ 0; +1, −4, +3⟩ ⟨ 0; +1, −3, +2⟩ ⟨ 0; +1, −2, +1⟩ ⟨ 0; +1, −1, 0⟩ ⟨ 0; +1, 0, −1⟩ ⟨ 0; +1, +1, −5⟩ ⟨ 0; +1, +1, −4⟩ ⟨ 0; +1, +1, −3⟩ ⟨ 0; +1, +1, −2⟩ ⟨ 0; +1, +1, −1⟩ ⟨ 0; +1, +1, 0⟩ ⟨ 0; +1, +1, +1⟩ ⟨ 0; +1, +1, +2⟩ ⟨ 0; +1, +1, +3⟩ ⟨ 0; +1, +1, +4⟩ ⟨ 0; +1, +1, +5⟩ ⟨ 0; +1, +2, −3⟩ ⟨ 0; +1, +3, −4⟩ ⟨ 0; +1, +4, −5⟩
 ⟨ 0; +2, −5, +3⟩ ⟨ 0; +2, −3, +1⟩ ⟨ 0; +2, −1, −1⟩ ⟨ 0; +2, +1, −3⟩ ⟨ 0; +2, +2, −5⟩ ⟨ 0; +2, +2, −3⟩ ⟨ 0; +2, +2, −1⟩ ⟨ 0; +2, +2, +1⟩ ⟨ 0; +2, +2, +3⟩ ⟨ 0; +2, +2, +5⟩ ⟨ 0; +2, +3, −5⟩ ⟨ 0; +3, −5, +2⟩ ⟨ 0; +3, −4, +1⟩ ⟨ 0; +3, −2, −1⟩ ⟨ 0; +3, −1, −2⟩ ⟨ 0; +3, +1, −4⟩ ⟨ 0; +3, +2, −5⟩ ⟨ 0; +3, +3, −5⟩ ⟨ 0; +3, +3, −4⟩ ⟨ 0; +3, +3, −2⟩ ⟨ 0; +3, +3, −1⟩ ⟨ 0; +3, +3, +1⟩ ⟨ 0; +3, +3, +2⟩ ⟨ 0; +3, +3, +4⟩ ⟨ 0; +3, +3, +5⟩
 ⟨ 0; +4, −5, +1⟩ ⟨ 0; +4, −3, −1⟩ ⟨ 0; +4, −1, −3⟩ ⟨ 0; +4, +1, −5⟩ ⟨ 0; +4, +4, −5⟩ ⟨ 0; +4, +4, −3⟩ ⟨ 0; +4, +4, −1⟩ ⟨ 0; +4, +4, +1⟩ ⟨ 0; +4, +4, +3⟩ ⟨ 0; +4, +4, +5⟩ ⟨ 0; +5, −4, −1⟩ ⟨ 0; +5, −3, −2⟩ ⟨ 0; +5, −2, −3⟩ ⟨ 0; +5, −1, −4⟩ ⟨ 0; +5, +5, −4⟩ ⟨ 0; +5, +5, −3⟩ ⟨ 0; +5, +5, −2⟩ ⟨ 0; +5, +5, −1⟩ ⟨ 0; +5, +5, +1⟩ ⟨ 0; +5, +5, +2⟩ ⟨ 0; +5, +5, +3⟩ ⟨ 0; +5, +5, +4⟩