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 commutative and associative:
d × e = e × d
(d × e) × f = d × (e × f)
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.
There is not necessarily an identity for multiplication.
Multiplicative rotativity is much like its additive counterpart:
d⊐ × e⊐ = (d × e)⊐
The product of two atoms is not necessarily an atom, but will be a linear combination of atoms.
With the addition and multiplication has defined here, each algebra of family F+ is the kind of ring that does not necessarily have a multiplicative identity.
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 |
An examination follows.
Let p, q, r be integer constants. If:
a × a = p × a + q × b + r × c = ⟨ p, q, r ⟩
then because of rotativity:
a⊐ × a⊐ = p × a⊐ + q × b⊐ + r × c⊐
whence:
b × b = p × b + q × c + r × a = ⟨ r, p, q ⟩
Similarly, c × c = ⟨ q, r, p ⟩.
Let t, u, v be integer constants. If:
a × b = t × a + u × b + v × c = ⟨ t, u, v ⟩
then by rotativity b × c = ⟨ v, t, u ⟩ and c × a = ⟨ u, v, t ⟩.
Along the same lines, let x, y, z be integer constants. If:
a × c = x × a + y × b + z × c = ⟨ x, y, z ⟩
then again by rotativity b × a = ⟨ z, x, y ⟩ and c × b = ⟨ y, z, x ⟩.
Now because of commutativity, the following are equal:
a × b = ⟨ t, u, v ⟩
b × a = ⟨ z, x, y ⟩
meaning that z = t, x = u, y = v. The same result would have been obtained from the equations b × c = c × b or c × a = a × c.
To summarize:
| a × a = ⟨ p, q, r ⟩ | b × a = ⟨ t, u, v ⟩ | c × a = ⟨ u, v, t ⟩ | ||
| a × b = ⟨ t, u, v ⟩ | b × b = ⟨ r, p, q ⟩ | c × b = ⟨ v, t, u ⟩ | ||
| a × c = ⟨ u, v, t ⟩ | b × c = ⟨ v, t, u ⟩ | c × c = ⟨ q, r, p ⟩ |
Incidentally,
(a × c)⊐ = a × b
(b × a)⊐ = b × c
(c × b)⊐ = c × a
Convenient to characterize a multiplication is its script, which is an ordered sextuple. In the middle is a semicolon for ease of reading:
⟨ p, q, r; t, u, v ⟩
In full:
⟨ (a × a)a, (a × a)b, (a × a)c; (a × b)a, (a × b)b, (a × b)c ⟩
In brief:
⟨ a × a; 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 ⟨ +1, +1, +1; 0, 0, −1 ⟩:
| a × a | = ⟨ | +1, | +1, | +1 ⟩ | b × a | = ⟨ | 0, | 0, | −1 ⟩ | c × a | = ⟨ | 0, | −1, | 0 ⟩ | ||
| a × b | = ⟨ | 0, | 0, | −1 ⟩ | b × b | = ⟨ | +1, | +1, | +1 ⟩ | c × b | = ⟨ | −1, | 0, | 0 ⟩ | ||
| a × c | = ⟨ | 0, | −1, | 0 ⟩ | b × c | = ⟨ | −1, | 0, | 0 ⟩ | c × c | = ⟨ | +1, | +1, | +1 ⟩ |
More compactly:
a × a = b × b = c × c = a + b + c
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: ⟨ −1, −1, −1; 0, 0, +1 ⟩.
Applying a constant factor to all the products yields a valid but different multiplication — and consequently a different algebra. Precisely 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 | = ⟨ | +5, | +5, | +5 ⟩ | b × a | = ⟨ | 0, | 0, | −5 ⟩ | c × a | = ⟨ | 0, | −5, | 0 ⟩ | ||
| a × b | = ⟨ | 0, | 0, | −5 ⟩ | b × b | = ⟨ | +5, | +5, | +5 ⟩ | c × b | = ⟨ | −5, | 0, | 0 ⟩ | ||
| a × c | = ⟨ | 0, | −5, | 0 ⟩ | b × c | = ⟨ | −5, | 0, | 0 ⟩ | c × c | = ⟨ | +5, | +5, | +5 ⟩ |
If that constant factor happens to be zero, the result becomes trivial. Four trivial F+ multiplications.
Here are four examples of non-trivial multiplications:
Script ⟨ +2, +1, +2; −1, 0, −1 ⟩:
| a × a | = ⟨ | +2, | +1, | +2 ⟩ | b × a | = ⟨ | −1, | 0, | −1 ⟩ | c × a | = ⟨ | 0, | −1, | −1 ⟩ | ||
| a × b | = ⟨ | −1, | 0, | −1 ⟩ | b × b | = ⟨ | +2, | +2, | +1 ⟩ | c × b | = ⟨ | −1, | −1, | 0 ⟩ | ||
| a × c | = ⟨ | 0, | −1, | −1 ⟩ | b × c | = ⟨ | −1, | −1, | 0 ⟩ | c × c | = ⟨ | +1, | +2, | +2 ⟩ |
Script ⟨ +4, +4, +2; +1, −1, −3 ⟩:
| a × a | = ⟨ | +4, | +4, | +2 ⟩ | b × a | = ⟨ | +1, | −1, | −3 ⟩ | c × a | = ⟨ | −1, | −3, | +1 ⟩ | ||
| a × b | = ⟨ | +1, | −1, | −3 ⟩ | b × b | = ⟨ | +2, | +4, | +4 ⟩ | c × b | = ⟨ | −3, | +1, | −1 ⟩ | ||
| a × c | = ⟨ | −1, | −3, | +1 ⟩ | b × c | = ⟨ | −3, | +1, | −1 ⟩ | c × c | = ⟨ | +4, | +2, | +4 ⟩ |
Script ⟨ +5, +2, +4; 0, +2, −4 ⟩:
| a × a | = ⟨ | +5, | +2, | +4 ⟩ | b × a | = ⟨ | 0, | +2, | −4 ⟩ | c × a | = ⟨ | +2, | −4, | 0 ⟩ | ||
| a × b | = ⟨ | 0, | +2, | −4 ⟩ | b × b | = ⟨ | +4, | +5, | +2 ⟩ | c × b | = ⟨ | −4, | 0, | +2 ⟩ | ||
| a × c | = ⟨ | +2, | −4, | 0 ⟩ | b × c | = ⟨ | −4, | 0, | +2 ⟩ | c × c | = ⟨ | +2, | +4, | +5 ⟩ |
Script ⟨ +7, +1, +5; −2, +2, −3 ⟩:
| a × a | = ⟨ | +7, | +1, | +5 ⟩ | b × a | = ⟨ | −2, | +2, | −3 ⟩ | c × a | = ⟨ | +2, | −3, | −2 ⟩ | ||
| a × b | = ⟨ | −2, | +2, | −3 ⟩ | b × b | = ⟨ | +5, | +7, | +1 ⟩ | c × b | = ⟨ | −3, | −2, | +2 ⟩ | ||
| a × c | = ⟨ | +2, | −3, | −2 ⟩ | b × c | = ⟨ | −3, | −2, | +2 ⟩ | c × c | = ⟨ | +1, | +5, | +7 ⟩ |
If ⟨ p, q, r; t, u, v ⟩ gives a valid multiplication, so does ⟨ p, r, q; u, t, v ⟩.
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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