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:
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