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}, 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,
Ei Ej. 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, Ei Ej,
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 right hand side the cordisan
E'i returns the image of Ej
as the product of Ei and Ej.
Since, versors are linear representations of verses (vectors in the vector space) the two are considered equivalent and will be referenced interchangeably. In particular, the versors Ei are the standard basis vectors (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1), each with a 1 in the ith position and zeros everywhere else, for 1 ≤ i ≤ 4.
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).