An algebra with ternary multiplication in a four-dimensional space.
Version of 18 March 2010.
Dave Barber's other pages.

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:

Associativity can be verified by elaborating the 45 = 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:

• From #I, we can exchange 1st-2nd, 1st-3rd and 2nd-3rd.
• From #II, we can exchange 2nd-3rd, 2nd-4th and 3rd-4th.
• From #III, we can exchange 3rd-4th, 3rd-5th and 4th-5th.
That leaves these exchanges:
• 1st-4th, which is effected by the sequence 1st-3rd, 3rd-4th, 1st-3rd.
• 1st-5th, which is effected by the sequence 1st-3rd, 3rd-5th, 1st-3rd.
• 2nd-5th, which is effected by the sequence 2nd-4th, 4th-5th, 2nd-4th.

We can subscript mulpos to emphasize the number of factors. For example, table one defined mulpos3, and shortly after that mulpos5 was introduced. One would naturally define mulpos1 as an identity operation. What mulposn 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:

 mulpos5 (mulpos3 (p, q, r), s, t, u, v) #IV mulpos5 (p, mulpos3 (q, r, s), t, u, v) #V mulpos5 (p, q, mulpos3 (r, s, t), u, v) #VI mulpos5 (p, q, r, mulpos3 (s, t, u), v) #VII mulpos5 (p, q, r, s, mulpos3 (t, u, v)) #VIII mulpos3 (mulpos5 (p, q, r, s, t), u, v) #IX mulpos3 (p, mulpos5 (q, r, s, t, u), v) #X mulpos3 (p, q, mulpos5 (r, s, t, u, v)) #XI

Looking at #IV through #VIII, we might say that mulpos5 is associative over mulpos3, while #IX through #XI suggest saying that mulpos3 is associative over mulpos5. We can write mulpos7 (p, q, r, s, t, u, v) without ambiguity because all eight decompositions are equal.

Commutativity of mulpos7 can be verified by exchanges similar to those used for mulpos5 above.

Analogous are mulpos9, mulpos11 et cetera. Like mulpos5, all these are associative and commutative.

If two factors to an invocation of mulposn are equal, they may be eliminated, leaving an invocation of mulposn−2. For example, consider:

mulpos7 (b, d, a, c, c, a, d)

By commutativity, this equals:

mulpos7 (a, a, b, c, c, d, d)

By associativity, this becomes:

mulpos3 (a, a, mulpos5 (b, c, c, d, d))

From table one, this equals:

mulpos5 (b, c, c, d, d)

Further reductions yield:

mulpos3 (b, c, c)
mulpos1 (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 mulposn, they may be removed leaving an invocation of mulposn−4. For instance, we may delete the underscored factors from this:

mulpos7 (b, c, c, d, a, d, d)

leaving:

mulpos3 (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 pq. (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:

e1, f1, g1, h1 ⟩ + ⟨ e2, f2, g2, h2 ⟩ = ⟨ e1 + e2, f1 + f2, g1 + g2, h1 + h2

Multiplication by a scalar n is useful:

ne, 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 TU.

The four components of T are written Ta, Tb, Tc and Td respectively; this leads to a terse notational tautology T = ⟨ Ta, Tb, Tc, Td ⟩.

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 mulnegn correspond to those for mulposn. Similarly, mulnegn 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 mulnegn, and the four atoms are removed leaving an invocation of mulnegn−4, the value of the invocation is negated. Here is an example, where again the atoms to be removed are underscored:

 mulpos7 (b, c, c, d, a, d, d) = +mulpos3 (c, d, d) = +c mulneg7 (b, c, c, d, a, d, d) = −mulneg3 (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 = mulpos3 (T, U, V) take the plus-or-minus signs to be plus, and to obtain W = mulneg3 (T, U, V) take the plus-or-minus signs to be minus.

Wa = TaUbVb + TaUcVc + TaUdVd + UaVbTb + UaVcTc + UaVdTd + VaTbUb + VaTcUc + VaTdUd
+ TaUaVa ± (TbUcVd + TcUdVb + TdUbVc + TbUdVc + TcUbVd + TdUcVb)

Wb = TbUcVc + TbUdVd + TbUaVa + UbVcTc + UbVdTd + UbVaTa + VbTcUc + VbTdUd + VbTaUa
+ TbUbVb ± (TcUdVa + TdUaVc + TaUcVd + TcUaVd + TdUcVa + TaUdVc)

Wc = TcUdVd + TcUaVa + TcUbVb + UcVdTd + UcVaTa + UcVbTb + VcTdUd + VcTaUa + VcTbUb
+ TcUcVc ± (TdUaVb + TaUbVd + TbUdVa + TdUbVa + TaUdVb + TbUaVd)

Wd = TdUaVa + TdUbVb + TdUcVc + UdVaTa + UdVbTb + UdVcTc + VdTaUa + VdTbUb + VdTcUc
+ TdUdVd ± (TaUbVc + TbUcVa + TcUaVb + TaUcVb + TbUaVc + TcUbVa)

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 mulpos2 (a, a). Postulating full associativity and commutativity, we can write:

mulpos3 (a, a, p) = mulpos2 (mulpos2 (a, a), p) = mulpos2 (e, p) = p

Along the same lines, define

mulpos (b, b) = mulpos (c, c) = mulpos (d, d) = e

With mulpos3 (a, b, p) equalling mulpos3 (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:

mulpos2 (a, e) = mulpos2 (a, mulpos2 (a, a)) = mulpos3 (a, a, a) = a.

with similar results for b, c and d. Also,

mulpos2 (a, f) = mulpos2 (a, mulpos2 (a, b)) = mulpos3 (a, a, b) = b.

By the same token, mulpos (a, g) = c, mulpos (a, h) = d, mulpos (b, f) = a et cetera. Further,

mulpos2 (f, g) = mulpos2 (mulpos2 (a, b), mulpos2 (a, c)) = mulpos4 (a, b, a, c)
= mulpos4 (a, a, b, c) = mulpos2 (mulpos2 (a, a), mulpos2 (b, c)) = mulpos2 (e, h) = h

Such line of reasoning allows completion the mulpos2 table:

mulpos2 abcdefgh
a efghabcd