ALGEBRA

The linear representations of verses (versors) were obtained through the applications of the left and right signed weighted interval (SWI) operators Li and Ri which led to a set of linear systems of equations whose solutions were the vector representations of the verses in the vector space. Accordingly, a given verse can have several versors.

In a vector space two types of operations can be performed on vectors: addition (form an abelian group) and scalar multiplication where the vectors can be scaled by elements from a fixed ground field, e.g. magnifying or shrinking a vector via multiplication by numbers. However, enhancing a vector space with a product operation that turns it into an algebra can significantly increase its utility. Introducing a product allows to multiply vectors, enabling richer algebraic structures and applications. The product (denoted by concatenation of vectors) as usual distributes over addition on both sides:

(U + V)W = UW + VW
U(V + W) = UV + UW,

for all vectors U, V and W in the vector space.

The choice of a basis for the vector space as before is {1:1, 1:7, 9:1, 9:127}, with the verses in their natural order (ordered basis), also conveniently denoted as {E1, E2, E3, E4}. Since it consists of beginning and ending verses of the two chapters one and nine the algebra will be referred to as (co)bif-19. Given two arbitrary vectors U and V, they can be expanded in terms of the basis vectors:

U = ∑ ci Ei,   for 1 ≤ i ≤ 4
V = ∑ dj Ej,   for 1 ≤ j ≤ 4,

with coefficients ci and dj in F12697. It follows that their product,

UV = ∑ ∑ ci dj Ei Ej,   for 1 ≤ i,j ≤ 4,

is uniquely determined in terms of the products of the basis vectors, EiEj. Hence, it is sufficient to only define the multiplication on the basis vectors which would then extend linearly to all vectors in the algebra.

To define the products of the basis vectors, EiEj, it is instructive to review the usual multiplication of ordinary numbers. The multiplication of numbers can be thought of as a rule or a function of two variables, M(⋅, ⋅), taking in two numbers and outputting one number, their product:

M(x, y) = xy.

A function of two variables can equivalently be expressed as a sequence of functions of one variable:

M(x, y) ≡ Px(y),

where,

Px(y) = xy,

is the sequence of functions indexed by x, taking one input, y, and returning its product with x. Stated differently, the one-variable function Px(y) can be thought of as a function that is attached to number x. It takes a number y as input and returns the product xy. This method of converting a function with multiple variables into a sequence of functions of a single variable is called currying. It treats the first variable as a parameter to index the sequence of functions taking one fewer variable.

Similarly, let M(Ei, Ej) be the multiplication function to be defined on the basis vectors. Then, via currying it transforms into a sequence of one-variable functions, PEi(Ej), ranging over the four basis vectors, Ei, taking Ej as input and returning the product Ei Ej as output. PEi(⋅) is the function that is attached to the basis vector Ei that takes a vector (represented by "⋅") and returns another vector.

Since, cordisans form a sequence of one-variable functions indexed by verses (each verse has its own cordisan), they are the ideal candidates to define the product:

PEi(⋅) = E'i(⋅),   (1 ≤ i ≤ 4),

where E'i is the cordisan of Ei. Putting them all together leads to the following definition:

Ei Ej = M(Ei, Ej) = PEi(Ej) = E'i(Ej),   (1 ≤ i,j ≤ 4),

where on the RHS the cordisan E'i returns the image of Ej as the product of Ei and Ej. Equipped with this binary operation the set of verses forms a magma.

With the multiplication so defined, the left/right signed interval operators SWI take a simpler form. Recall, they were defined as Li = ∥[Ei, V] O∥ and Ri = ∥O [Ei, V]∥, for 1 ≤ i ≤ 4, with O as the orconion. The action of a verse U on O results in its cordisan U' which in turn acts on another verse V, (U'V), that by definition is the product UV. Viewed this way, the action by Li is simply the left multiplication by ∥[Ei, V]∥ and similarly that of Ri is the right multiplication by the same interval operator ∥[Ei, V]∥. Thus, the left and right SWI operators, Li and Ri, unify into a single signed interval operator SI with both left and right actions. The actions of the SI operator quantify the interactions between the basis verses Ei and a verse V, allowing the derivation of the linear representation of the verse (its versor) as shown previously.

Since, versors are linear representations of verses (vectors in the vector space), a verse and its versor are considered equivalent and will be referenced interchangeably. The versors of the basis verses Ei can readily be obtained:

E1 = 1 E1 + 0 E2 + 0 E3 + 0 E4 = (1, 0, 0, 0)
E2 = 0 E1 + 1 E2 + 0 E3 + 0 E4 = (0, 1, 0, 0)
E3 = 0 E1 + 0 E2 + 1 E3 + 0 E4 = (0, 0, 1, 0)
E4 = 0 E1 + 0 E2 + 0 E3 + 1 E4 = (0, 0, 0, 1)

The versor of a basis verse Ei has 1 in the ith position and zeros elsewhere. They form the standard basis.

The product of two basis vectors is another vector, not necessarily a basis vector. Therefore, it, too, can be expressed as a linear combination of the basis vectors:

Ei Ej = ∑ cijk Ek,   for 1 ≤ i,j,k ≤ 4,

where, the product Ei Ej itself is expressed as a linear combination of basis vectors Ek. Hence, the products of basis vectors are defined by a multiplication table consisting of the coefficient cijk also know as the structure constants. The multiplication then extends linearly to all vectors in the algebra. There are a total of 64, (= 4 × 4 × 4), structure constants for the chosen basis of length four.

The product of a pair of basis vectors (basis verses) Ei Ej in general is not a basis verse. A given verse has several associated versors: two versors from the left SWI operator Li due to the presence of singleton intervals of signed lengths ± 1 (one for each sign) and one versor from the right SWI Ri. A total of three versors are thus associated with a verse. Each such versor will lead to a distinct multiplication table and hence a separate algebra. The three (co)bif-19 algebras are accordingly labeled with types (L+), (L−) and (R).

The followings are the three (co)bif-19 algebras over F12697.

(L+):

E1 E1 = (1 0 0 0) = E1
E1 E2 = (-2909, -3377, 5372, -5867)
E1 E3 = (673, -5777, 2006, 300)
E1 E4 = (-2893, -4362, -1974, 4319)

E2 E1 = (1 0 0 0) = E1
E2 E2 = (1341, 2575, 131, 5952)
E2 E3 = (1796, -335, -2541, -173)
E2 E4 = (-5405, 3093, 2124, -5590)

E3 E1 = (1 0 0 0) = E1
E3 E2 = (-3341, -1141, 2923, -765)
E3 E3 = (-6092, -3529, -3238, -5444)
E3 E4 = (-415, -3344, -4184, -1792)

E4 E1 = (1 0 0 0) = E1
E4 E2 = (4344, 1920, -5301, 5564)
E4 E3 = (4792, -5223, 5553, 3634)
E4 E4 = (375, 1584, -4703, 5204)

(L−):

E1 E1 = (1 0 0 0) = E1
E1 E2 = (2909, -3377, 5372, -5867)
E1 E3 = (-673, -5777, 2006, 300)
E1 E4 = (2893, -4362, -1974, 4319)

E2 E1 = (1 0 0 0) = E1
E2 E2 = (-1341, 2575, 131, 5952)
E2 E3 = (-1796, -335, -2541, -173)
E2 E4 = (5405, 3093, 2124, -5590)

E3 E1 = (1 0 0 0) = E1
E3 E2 = (3341, -1141, 2923, -765)
E3 E3 = (6092, -3529, -3238, -5444)
E3 E4 = (415, -3344, -4184, -1792)

E4 E1 = (1 0 0 0) = E1
E4 E2 = (-4344, 1920, -5301, 5564)
E4 E3 = (-4792, -5223, 5553, 3634)
E4 E4 = (-375, 1584, -4703, 5204)

Note the two (Left) multiplication tables differ only in their first components with opposite signs while their remaining three components are the same. This is due to basis verse E1 being the fixed point of all cordisans, resulting in a directed singleton interval.

(R):

E1 E1 = (1 0 0 0) = E1
E1 E2 = (11, 1401, -1200, -3285)
E1 E3 = (-2927, -3876, 1302, 67)
E1 E4 = (2424, 2097, -5774, -4076)

E2 E1 = (1 0 0 0) = E1
E2 E2 = (2992, -1736, 386, 3487)
E2 E3 = (-4871, -6169, 2173, 5930)
E2 E4 = (137, 5392, 3550, 3967)

E3 E1 = (1 0 0 0) = E1
E3 E2 = (-2476, 5182, 1922, 2855)
E3 E3 = (4265, -100, -3609, -2824)
E3 E4 = (2498, 3917, -475, 2885)

E4 E1 = (1 0 0 0) = E1
E4 E2 = (-3186, -2455, -6322, 3467)
E4 E3 = (-2489, -5021, 2158, 926)
E4 E4 = (1068, -5651, 5450, -5392)

The products Ei Ej are expressed on the right hand sides as vectors whose components are the structure constants cijk. They can also be expressed in fully expended forms in terms of the basis vectors. For instance, the last equation will take the following form when fully expanded:

E4 E4 = 1068 E1 − 5651 E2 + 5450 E3 − 5392 E4.

The exceptional status of E1 (versor of 1:1) is readily apparent in the table:

Ei E1 = (1 0 0 0) = E1,   for 1 ≤ i ≤ 4.

This is because 1:1 (and hence, E1) is a fixed point of all cordisans (the only such verse). E1, therefore, is a right-absorbing element among the basis vectors. In other words, it behaves like zero when it is multiplied from the right. However, it behaves similar to other basis vectors when it appears on the left. A consequence of this exception is the presence of zero divisors; nonzero vectors whose product is zero, unlike ordinary numbers, as illustrated below. The two nonzero vectors V = E2 − E3 and E1 have zero product:

V E1 = (E2 − E3) E1 = E2 E1 − E3 E1 = E1 − E1 = 0.

However, the product E1 V is a nonzero vector. From the multiplication tables it is readily apparent that the vector product is not commutative. The following example from (co)bif-19 (R) algebra shows it is neither associative:

(E1E2)E3 = (-1700, 5652, -4263, -3797)
E1(E2E3) = (5594, 4298, 2253, -1612).