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:

• No number has an absolute value greater than nine.
• The numbers have no common factor greater than one.
• The first number is not negative.
• a × bb × a.

 ⟨ +1, +1, +1; 0, 0, −1⟩ ⟨ +2, 0, 0; +1, +1, −1⟩ ⟨ +2, 0, +1; 0, +1, −1⟩ ⟨ +2, +1, 0; +1, 0, −1⟩ ⟨ +2, +1, +2; −1, 0, −1⟩ ⟨ +2, +2, +1; 0, −1, −1⟩ ⟨ +4, −1, 0; +1, +2, −1⟩ ⟨ +4, 0, −1; +2, +1, −1⟩ ⟨ +4, 0, +2; −1, +1, −1⟩ ⟨ +4, +1, +2; +1, +2, −3⟩ ⟨ +4, +2, 0; +1, −1, −1⟩ ⟨ +4, +2, +1; +2, +1, −3⟩ ⟨ +4, +2, +3; −2, −1, −1⟩ ⟨ +4, +2, +4; −1, +1, −3⟩ ⟨ +4, +3, +2; −1, −2, −1⟩ ⟨ +4, +4, +2; +1, −1, −3⟩ ⟨ +4, +4, +5; −2, −1, −3⟩ ⟨ +4, +5, +4; −1, −2, −3⟩
 ⟨ +5, −1, −1; +2, +2, −1⟩ ⟨ +5, −1, +1; 0, +2, −1⟩ ⟨ +5, +1, −1; +2, 0, −1⟩ ⟨ +5, +1, +3; −2, 0, −1⟩ ⟨ +5, +2, +2; +2, +2, −4⟩ ⟨ +5, +2, +4; 0, +2, −4⟩ ⟨ +5, +3, +1; 0, −2, −1⟩ ⟨ +5, +3, +3; −2, −2, −1⟩ ⟨ +5, +4, +2; +2, 0, −4⟩ ⟨ +5, +4, +6; −2, 0, −4⟩ ⟨ +5, +6, +4; 0, −2, −4⟩ ⟨ +5, +6, +6; −2, −2, −4⟩ ⟨ +7, −1, +1; +2, +4, −3⟩ ⟨ +7, 0, +2; +2, +4, −4⟩ ⟨ +7, +1, −1; +4, +2, −3⟩ ⟨ +7, +1, +5; −2, +2, −3⟩ ⟨ +7, +2, 0; +4, +2, −4⟩ ⟨ +7, +2, +6; −2, +2, −4⟩ ⟨ +7, +5, +1; +2, −2, −3⟩ ⟨ +7, +5, +7; −4, −2, −3⟩ ⟨ +7, +6, +2; +2, −2, −4⟩ ⟨ +7, +6, +8; −4, −2, −4⟩ ⟨ +7, +7, +5; −2, −4, −3⟩ ⟨ +7, +8, +6; −2, −4, −4⟩
 ⟨ +8, −2, −1; +2, +3, −1⟩ ⟨ +8, −2, 0; +1, +3, −1⟩ ⟨ +8, −1, −2; +3, +2, −1⟩ ⟨ +8, −1, +2; −1, +2, −1⟩ ⟨ +8, 0, −2; +3, +1, −1⟩ ⟨ +8, 0, +3; −2, +1, −1⟩ ⟨ +8, +2, −1; +2, −1, −1⟩ ⟨ +8, +2, +4; −3, −1, −1⟩ ⟨ +8, +3, 0; +1, −2, −1⟩ ⟨ +8, +3, +4; −3, −2, −1⟩ ⟨ +8, +4, +2; −1, −3, −1⟩ ⟨ +8, +4, +3; −2, −3, −1⟩ ⟨ +8, +4, +5; +2, +3, −7⟩ ⟨ +8, +4, +6; +1, +3, −7⟩ ⟨ +8, +5, +4; +3, +2, −7⟩ ⟨ +8, +5, +8; −1, +2, −7⟩ ⟨ +8, +6, +4; +3, +1, −7⟩ ⟨ +8, +6, +9; −2, +1, −7⟩ ⟨ +8, +8, +5; +2, −1, −7⟩ ⟨ +8, +9, +6; +1, −2, −7⟩