Section 1. We begin by defining four atoms, namely a, b, c and d, and a ternary operation called mulpos. This is short for multiplication positive, which name will be justified later; the arguments to mulpos will correspondingly be called factors. Table one contains the details:
Table one: definition of mulpos | ||||
---|---|---|---|---|
mulpos (a, a, a) = a | mulpos (b, b, a) = a | mulpos (c, c, a) = a | mulpos (d, d, a) = a | mulpos (b, c, d) = a |
mulpos (a, a, b) = b | mulpos (b, b, b) = b | mulpos (c, c, b) = b | mulpos (d, d, b) = b | mulpos (a, c, d) = b |
mulpos (a, a, c) = c | mulpos (b, b, c) = c | mulpos (c, c, c) = c | mulpos (d, d, c) = c | mulpos (a, b, d) = c |
mulpos (a, a, d) = d | mulpos (b, b, d) = d | mulpos (c, c, d) = d | mulpos (d, d, d) = d | mulpos (a, b, c) = d |
mulpos is commutative, so that mulpos (a, b, c) = mulpos (a, c, b) = mulpos (c, a, b) et cetera |
If two of the factors are equal, the value of mulpos is equal to the other factor, as in the first four columns. If instead all factors are unequal, the value of mulpos is equal to whichever of a, b, c and d is not a factor, as in the fifth column.
Let p, q, r, s and t be any atoms. Then mulpos is associative, in this sense that these three expressions are equal:
mulpos (mulpos (p, q, r), s, t) | #I | |
mulpos (p, mulpos (q, r, s), t) | #II | |
mulpos (p, q, mulpos (r, s, t)) | #III |
Associativity can be verified by elaborating the 4^{5} = 1024 possibilities, for which a computer is helpful. Hence we can write the five-factor mulpos (p, q, r, s, t) as equalling any of these three decompositions.
Further, mulpos (p, q, r, s, t) is commutative, because the factors in these positions are pairwise exchangeable:
We can subscript mulpos to emphasize the number of factors. For example, table one defined mulpos_{3}, and shortly after that mulpos_{5} was introduced. One would naturally define mulpos_{1} as an identity operation. What mulpos_{n} might mean when n is even will be addressed later.
Let u and v be two more atoms. In an extension of the definition of associativity, the following eight expressions can be shown equal by means of a shifting technique:
mulpos_{5} (mulpos_{3} (p, q, r), s, t, u, v) | #IV | |
mulpos_{5} (p, mulpos_{3} (q, r, s), t, u, v) | #V | |
mulpos_{5} (p, q, mulpos_{3} (r, s, t), u, v) | #VI | |
mulpos_{5} (p, q, r, mulpos_{3} (s, t, u), v) | #VII | |
mulpos_{5} (p, q, r, s, mulpos_{3} (t, u, v)) | #VIII | |
mulpos_{3} (mulpos_{5} (p, q, r, s, t), u, v) | #IX | |
mulpos_{3} (p, mulpos_{5} (q, r, s, t, u), v) | #X | |
mulpos_{3} (p, q, mulpos_{5} (r, s, t, u, v)) | #XI |
Looking at #IV through #VIII, we might say that mulpos_{5} is associative over mulpos_{3}, while #IX through #XI suggest saying that mulpos_{3} is associative over mulpos_{5}. We can write mulpos_{7} (p, q, r, s, t, u, v) without ambiguity because all eight decompositions are equal.
Commutativity of mulpos_{7} can be verified by exchanges similar to those used for mulpos_{5} above.
Analogous are mulpos_{9}, mulpos_{11} et cetera. Like mulpos_{5}, all these are associative and commutative.
If two factors to an invocation of mulpos_{n} are equal, they may be eliminated, leaving an invocation of mulpos_{n−2}. For example, consider:
mulpos_{7} (b, d, a, c, c, a, d)
By commutativity, this equals:
mulpos_{7} (a, a, b, c, c, d, d)
By associativity, this becomes:
mulpos_{3} (a, a, mulpos_{5} (b, c, c, d, d))
From table one, this equals:
mulpos_{5} (b, c, c, d, d)
Further reductions yield:
mulpos_{3} (b, c, c)
mulpos_{1} (b)
If n is at least 5, then such reduction is always possible when the factors are atoms, because there are only four different atoms to pick from. Later we will introduce factors that are more complicated than atoms, and reduction will not be assured when n is large. Even then, reduction will work when two factors happen to be equal.
A second type of reduction is possible. If all four of the atoms a, b, c and d are present among the factors of an invocation of mulpos_{n}, they may be removed leaving an invocation of mulpos_{n−4}. For instance, we may delete the underscored factors from this:
mulpos_{7} (b, c, c, d, a, d, d)
leaving:
mulpos_{3} (c, d, d)
Recall that mulpos (p, p, q) = q. Had instead mulpos (p, p, q) been defined to equal p, associativity would have failed because mulpos (a, a, b, b, b) could then be decomposed two unequal ways:
mulpos (a, a, mulpos (b, b, b))
= mulpos (a, a, b) = a
mulpos (mulpos (a, a, b), b, b)
= mulpos (a, b, b) = b
Section 2. Atoms may be added, although the sum is not generally an atom. This operation is binary, associative and commutative.
First of all, p can be written 1p. Then, for integers e and f, ep + fp is combined into (e + f)p. Unlike atoms can be added, but the expression cannot be simplified. Hence ep + fq, although valid, is primitive when p ≠ q. (This is reminiscent of an expression like 3 + 4i in complex numbers.)
The next step is to form a tetrad, written between shallow angle brackets. With integers g and h, the general tetrad becomes
ea + fb + gc + hd = ⟨ e, f, g, h ⟩
From associativity and commutativity, addition must be by parallel components:
⟨ e_{1}, f_{1}, g_{1}, h_{1} ⟩ + ⟨ e_{2}, f_{2}, g_{2}, h_{2} ⟩ = ⟨ e_{1} + e_{2}, f_{1} + f_{2}, g_{1} + g_{2}, h_{1} + h_{2} ⟩
Multiplication by a scalar n is useful:
n ⟨ e, f, g, h ⟩ = ⟨ ne, nf, ng, nh ⟩
For tetrads T and U, two customary simplifications of notation are performed: (−1)T becomes −T, and T + (−U) becomes T − U.
The four components of T are written T_{a}, T_{b}, T_{c} and T_{d} respectively; this leads to a terse notational tautology T = ⟨ T_{a}, T_{b}, T_{c}, T_{d} ⟩.
The following identities should not be overlooked:
a = ⟨ 1, 0, 0, 0 ⟩ |
b = ⟨ 0, 1, 0, 0 ⟩ |
c = ⟨ 0, 0, 1, 0 ⟩ |
d = ⟨ 0, 0, 0, 1 ⟩ |
So far, mulpos has been defined only for atomic factors. What about factors that are more generally tetrads? These definitions, where T, U, V and W are tetrads, take care of the matter:
mulpos (T + U, V, W) = mulpos (T, V, W) + mulpos (U, V, W) |
mulpos (T, U + V, W) = mulpos (T, U, W) + mulpos (T, V, W) |
mulpos (T, U, V + W) = mulpos (T, U, V) + mulpos (T, U, W) |
Applied recursively sufficiently many times, perhaps with the aid of multiplication by the scalar −1, these rules will produce invocations of mulpos with atomic factors.
Clearly, mulpos is distributive over addition; from that, the value of mulpos (T, U, V) must be proportional to each factor. This property justifies the multiplication part of the multiplication positive name. One way that this operation differs from the multiplications found in most other algebras is that there is no multiplicative identity; still there is an annihilator ⟨ 0, 0, 0, 0 ⟩.
Tetrads of the form ⟨ e, f, 0, 0 ⟩ form a subset closed under addition and mulpos.
Section 3. A counterpart to mulpos is multiplication negative:
Table two: definition of mulneg | ||||
---|---|---|---|---|
mulneg (a, a, a) = a | mulneg (b, b, a) = a | mulneg (c, c, a) = a | mulneg (d, d, a) = a | mulneg (b, c, d) = −a |
mulneg (a, a, b) = b | mulneg (b, b, b) = b | mulneg (c, c, b) = b | mulneg (d, d, b) = b | mulneg (a, c, d) = −b |
mulneg (a, a, c) = c | mulneg (b, b, c) = c | mulneg (c, c, c) = c | mulneg (d, d, c) = c | mulneg (a, b, d) = −c |
mulneg (a, a, d) = d | mulneg (b, b, d) = d | mulneg (c, c, d) = d | mulneg (d, d, d) = d | mulneg (a, b, c) = −d |
mulneg is commutative, so that mulneg (a, b, c) = mulneg (a, c, b) = mulneg (c, a, b) et cetera |
Note that mulneg differs from mulpos only when the three factors are unequal.
Associativity and commutativity theorems for mulneg_{n} correspond to those for mulpos_{n}. Similarly, mulneg_{n} is distributive over addition.
There is one difference to note in the reduction rules: when all four atoms are among the factors to an invocation of mulneg_{n}, and the four atoms are removed leaving an invocation of mulneg_{n−4}, the value of the invocation is negated. Here is an example, where again the atoms to be removed are underscored:
mulpos_{7} (b, c, c, d, a, d, d) = +mulpos_{3} (c, d, d) = +c |
mulneg_{7} (b, c, c, d, a, d, d) = −mulneg_{3} (c, d, d) = −c |
What follows is a lengthy elaboration of the definition of the multiplications for tetrads, in a form that might be convenient for programming a computer. To obtain W = mulpos_{3} (T, U, V) take the plus-or-minus signs to be plus, and to obtain W = mulneg_{3} (T, U, V) take the plus-or-minus signs to be minus.
W_{a} = T_{a}U_{b}V_{b}
+ T_{a}U_{c}V_{c} + T_{a}U_{d}V_{d}
+ U_{a}V_{b}T_{b} + U_{a}V_{c}T_{c}
+ U_{a}V_{d}T_{d} + V_{a}T_{b}U_{b}
+ V_{a}T_{c}U_{c} + V_{a}T_{d}U_{d}
+ T_{a}U_{a}V_{a}
± (T_{b}U_{c}V_{d} + T_{c}U_{d}V_{b}
+ T_{d}U_{b}V_{c} + T_{b}U_{d}V_{c}
+ T_{c}U_{b}V_{d} + T_{d}U_{c}V_{b})
W_{b} = T_{b}U_{c}V_{c}
+ T_{b}U_{d}V_{d} + T_{b}U_{a}V_{a}
+ U_{b}V_{c}T_{c} + U_{b}V_{d}T_{d}
+ U_{b}V_{a}T_{a} + V_{b}T_{c}U_{c}
+ V_{b}T_{d}U_{d} + V_{b}T_{a}U_{a}
+ T_{b}U_{b}V_{b}
± (T_{c}U_{d}V_{a} + T_{d}U_{a}V_{c}
+ T_{a}U_{c}V_{d} + T_{c}U_{a}V_{d}
+ T_{d}U_{c}V_{a} + T_{a}U_{d}V_{c})
W_{c} = T_{c}U_{d}V_{d}
+ T_{c}U_{a}V_{a} + T_{c}U_{b}V_{b}
+ U_{c}V_{d}T_{d} + U_{c}V_{a}T_{a}
+ U_{c}V_{b}T_{b} + V_{c}T_{d}U_{d}
+ V_{c}T_{a}U_{a} + V_{c}T_{b}U_{b}
+ T_{c}U_{c}V_{c}
± (T_{d}U_{a}V_{b} + T_{a}U_{b}V_{d}
+ T_{b}U_{d}V_{a} + T_{d}U_{b}V_{a}
+ T_{a}U_{d}V_{b} + T_{b}U_{a}V_{d})
W_{d} = T_{d}U_{a}V_{a}
+ T_{d}U_{b}V_{b} + T_{d}U_{c}V_{c}
+ U_{d}V_{a}T_{a} + U_{d}V_{b}T_{b}
+ U_{d}V_{c}T_{c} + V_{d}T_{a}U_{a}
+ V_{d}T_{b}U_{b} + V_{d}T_{c}U_{c}
+ T_{d}U_{d}V_{d}
± (T_{a}U_{b}V_{c} + T_{b}U_{c}V_{a}
+ T_{c}U_{a}V_{b} + T_{a}U_{c}V_{b}
+ T_{b}U_{a}V_{c} + T_{c}U_{b}V_{a})
With this definition, the components of a tetrad need not be limited to integers, but may easily be extended to the complex numbers.
Despite the similarity of their definitions, mulpos and mulneg can give substantially different results. For instance, mulpos (a + b, b + c, c + d) = ⟨ 2, 2, 2, 2 ⟩, but mulneg (a + b, b + c, c + d) = ⟨ 0, 0, 0, 0 ⟩.
Section 4. Multiplicative associativity and commutativity as described so far have been full, but weaker versions are available to anyone who is designing a ternary function. Let mul denote a variant of mulpos, and consider the following expressions:
mul (mul (p, q, r), s, t) | #I′ | |
mul (p, mul (q, r, s), t) | #II′ | |
mul (p, q, mul (r, s, t)) | #III′ |
Under alternate associativity, #I′ must equal #III′, but #II′ is permitted to be something different.
Under alternate commutativity, only this much is guaranteed:
mul (p, q, r) = mul (r, q, p) |
mul (q, r, p) = mul (p, r, q) |
mul (r, p, q) = mul (q, p, r) |
Under cyclical commutativity, only these are assured:
mul (p, q, r) = mul (q, r, p) = mul (r, p, q) |
mul (p, r, q) = mul (q, p, r) = mul (r, q, p) |
Observe that if mul is both alternatively and cyclically commutative, then it is fully commutative.
By way of example, mulim (for multiplication imaginary) is associative but not commutative. As usual, i represents a square root of −1.
Table three: definition of mulim | |||
---|---|---|---|
mulim (a, a, a) = +ia | mulim (b, b, b) = +b | mulim (c, c, c) = +c | mulim (d, d, d) = +id |
mulim (a, a, b) = +ib | mulim (b, b, c) = +c | mulim (c, c, d) = +d | mulim (d, d, a) = +ia |
mulim (a, b, a) = −ib | mulim (b, c, b) = +c | mulim (c, d, c) = +d | mulim (d, a, d) = −ia |
mulim (b, a, a) = +ib | mulim (c, b, b) = +c | mulim (d, c, c) = +d | mulim (a, d, d) = +ia |
mulim (a, a, c) = +ic | mulim (b, b, d) = +d | mulim (c, c, a) = +a | mulim (d, d, b) = +ib |
mulim (a, c, a) = +ic | mulim (b, d, b) = −d | mulim (c, a, c) = +a | mulim (d, b, d) = −ib |
mulim (c, a, a) = +ic | mulim (d, b, b) = +d | mulim (a, c, c) = +a | mulim (d, d, c) = +ic |
mulim (a, a, d) = +id | mulim (b, b, a) = +a | mulim (c, c, b) = +b | mulim (b, d, d) = +ib |
mulim (a, d, a) = −id | mulim (b, a, b) = −a | mulim (c, b, c) = +b | mulim (d, c, d) = +ic |
mulim (d, a, a) = +id | mulim (a, b, b) = +a | mulim (b, c, c) = +b | mulim (c, d, d) = +ic |
mulim (d, b, c) = −ia | mulim (a, c, d) = +b | mulim (b, d, a) = −c | mulim (c, a, b) = +id |
mulim (c, b, d) = +ia | mulim (d, c, a) = −b | mulim (a, d, b) = +c | mulim (b, a, c) = −id |
mulim (b, c, d) = +ia | mulim (c, d, a) = −b | mulim (d, a, b) = −c | mulim (a, b, c) = +id |
mulim (d, c, b) = −ia | mulim (a, d, c) = +b | mulim (b, a, d) = +c | mulim (c, b, a) = −id |
mulim (b, d, c) = +ia | mulim (c, a, d) = +b | mulim (d, b, a) = +c | mulim (a, c, b) = +id |
mulim (c, d, b) = −ia | mulim (d, a, c) = −b | mulim (a, b, d) = −c | mulim (b, c, a) = −id |
Many more variations of mulpos are found here.
Section 5. How might we define mulpos with an even number of arguments, for instance mulpos (a, b)? One approach is to admit four more atoms. We first establish e, which will equal mulpos_{2} (a, a). Postulating full associativity and commutativity, we can write:
mulpos_{3} (a, a, p) = mulpos_{2} (mulpos_{2} (a, a), p) = mulpos_{2} (e, p) = p
Along the same lines, define
mulpos (b, b) = mulpos (c, c) = mulpos (d, d) = e
With mulpos_{3} (a, b, p) equalling mulpos_{3} (c, d, p), we introduce the atom f:
mulpos (a, b) = mulpos (c, d) = f
Similarly create g and h:
mulpos (a, c) = mulpos (b, d) = g
mulpos (a, d) = mulpos (b, c) = h
Other combinations of arguments can now be resolved:
mulpos_{2} (a, e) = mulpos_{2} (a, mulpos_{2} (a, a)) = mulpos_{3} (a, a, a) = a.
with similar results for b, c and d. Also,
mulpos_{2} (a, f) = mulpos_{2} (a, mulpos_{2} (a, b)) = mulpos_{3} (a, a, b) = b.
By the same token, mulpos (a, g) = c, mulpos (a, h) = d, mulpos (b, f) = a et cetera. Further,
mulpos_{2} (f, g) =
mulpos_{2} (mulpos_{2} (a, b), mulpos_{2} (a, c)) =
mulpos_{4} (a, b, a, c)
=
mulpos_{4} (a, a, b, c) =
mulpos_{2} (mulpos_{2} (a, a), mulpos_{2} (b, c)) =
mulpos_{2} (e, h) = h
Such line of reasoning allows completion the mulpos_{2} table:
mulpos_{2} | a | b | c | d | e | f | g | h |
---|---|---|---|---|---|---|---|---|
a | e | f | g | h | a | b | c | d |
b | f | e | h | g | b | a | d | c |
c | g | h | e | f | c | d | a | b |
d | h | g | f | e | d | c | b | a |
e | a | b | c | d | e | f | g | h |
f | b | a | d | c | f | e | h | g |
g | c | d | a | b | g | h | e | f |
h | d | c | b | a | h | g | f | e |
This turns out to be a well-known abelian group of order eight, sometimes denoted Z_{2} × Z_{2} × Z_{2}.