LINEAR CONTENT

The constructions of the (co)bif-19 algebras were based entirely on the external properties of verses alone. They relied exclusively on the positions of verses relative to all other verses and chapters. A more complete and descriptive account of the properties of verses can be obtained if the internal properties of verses are taken into account as well. This can be achieved via the construction of algebras that combine the positions of verses in conjunction with their textual contents.

The textual content of a verse, (its literal composition), can be all or any appropriately selected subsets. Of particular interest are the initialed chapters where the initialed letters within verses readily form the appropriate subsets. Chp two is the first ALM-initialed chp. The ALM-content of its tetron consists of 3As, 4Ls, and 3Ms. Their sum, 3A + 4L + 3M, represents the linear content (LC) of the tetron. Note the similarity between the sum and a vector fully expanded as a linear combination of the basis vectors. Such a vector space can be attained if the three letters in the sum are taken to serve as the basis: B = {A, L, M}.

The letters A, L, M in the basis span a three-dimensional vector space where vectors are linear combinations of the three letters (the LCs): a A + l L + m M. The scalar coefficients a, l and m are in the same field F12697 as before. Next, the vector space will be turned into an algebra by defining a multiplication among LCs. This is achieved by defining a (co)bif algebra of the same dimension whose products on versors would carry over to the LC vector space, thus turning it into an algebra.

The basis for the (co)bif algebra will consist of three appropriately chosen verses containing the initialed letters. The distinguished set will consist of the following verses: the tetron, the declaration (the verse where the initial letters are first introduced), and the last verse. In other words, the header, the beginning and ending (the boundary) verses: {2:0, 2:1, 2:286}. For convenience, the basis will also be denoted as {E₁, E₂, E₃} as before. The applications of the left and right SWI operators will result in a pair of left/right (co)bif-2 algebras over F12697.

The followings are the multiplication tables for the left and right (co)bif-2 algebras over the same vector space:

(L):

E₁ E₁ = (5554, -4391, -1578)
E₁ E₂ = (412, 1543, 3772)
E₁ E₃ = (1919, -3001, -4542)

E₂ E₁ = (1030, -4440, -149)
E₂ E₂ = (-1923, -391, -5742)
E₂ E₃ = (526, -2365, -4338)

E₃ E₁ = (-415, 4337, 3157)
E₃ E₂ = (4135, 4551, -4181)
E₃ E₃ = (5125, 6183, 5576)

(R):

E₁ E₁ = (-2251, 1395, -2314)
E₁ E₂ = (-1235, -1722, -3819)
E₁ E₃ = (-3555, -146, 3274)

E₂ E₁ = (-5202, 3513, 284)
E₂ E₂ = (3074, -4580, 3696)
E₂ E₃ = (1188, 1829, 1208)

E₃ E₁ = (1810, -953, 2837)
E₃ E₂ = (-3784, -2887, 4665)
E₃ E₃ = (-4034, -4777, 511)

In the (co)bif-2 vector space (ignoring its algebra aspect) the vectors are linear combinations of the verses 2:0, 2:1, and 2:286, also denoted as E₁, E₂, E₃, respectively. In the LC vector space the vectors are linear combinations of the init-letters A, L and M, similarly denoted as F₁, F₂, F₃. The latter can be turned into an algebra via establishing a link between the two.

Let q be a bijective function from (co)bif algebra to the LC vector space. Then a vector product in the LC vector space can be defined with the help of q. Let U and V be vectors in LC. Since q is a bijection, there are unique vectors X and Y in (co)bif such that they are mapped to U and V under q:

U = q(X)
V = q(Y).

Since (co)bif is an algebra, XY, the product of X and Y, is defined. Under q, this product is mapped to some vector W in LC: q(XY) = W, which is then defined to be the product of U and V: UV = W. Putting it all together the multiplication in LC can be described as follows:

UV = q(q⁻¹(U) q⁻¹(V)),

where q⁻¹ is the inverse of q, it goes in the opposite direction from LC to (co)bif: X = q⁻¹(U) and Y = q⁻¹(V). This only works if q is a bijection where distinct vectors are mapped to distinct vectors. This method of defining an algebra on a vector space from an existing algebra is an example of a general construction called transport of structure.

The bijection q assigns to every versor (a vector in (co)bif vector space) a unique vector in LC, its linear content. Moreover, it preserves the addition and scalar multiplication among vectors: q(X + Y) = q(X) + q(Y) and q(cX) = cq(X), for vectors X and Y and the scalar c. It is a linear map. Since, q, in addition to being linear, it also preserves the vector multiplication, q(XY) = q(X)q(Y), (by definition), it is an algebra morphism. Morphisms between two algebras allow the algebraic operations on vectors in one algebra to be transferred/compared to those in another algebra. Since, q is a bijection, it is an isomorphism. Isomorphic algebras have equivalent algebraic structure.

It remains to define the bijective linear map q. This is a map that assigns linear contents to vectors. Let X be a vector in the left (co)bif-2 algebra. Then, it is a linear combinations of the basis vectors:

X = ∑ c_i E_i = c₁ E₁ + c₂ E₂ + c₃ E₃ = (c₁, c₂, c₃),

with the scalar coefficients c_i from F12697. Applying q to both sides gives:

q(X) = q(∑ c_i E_i) = q(c₁ E₁ + c₂ E₂ + c₃ E₃) = c₁ q(E₁) + c₂ q(E₂) + c₃ q(E₃).

The expression on the far right follows from linearity of q. It suffices to specify the action of q on the basis vectors E_i's. Then by linearity it extends to all vectors, uniquely determining q.

The linear contents of the basis vectors q(E_i)'s in fully expanded and in component forms are:

q(E₁) = q(2:0) = 3A + 4L + 3M = (3, 4, 3)
q(E₂) = q(2:1) = A + L + M = (1, 1, 1)
q(E₃) = q(2:286) = 48A + 24L + 11M = (48, 24, 11)

Replacing the linear contents for the q(E_i)'s on the right-hand-side (RHS) of q(X) gives:

q(X) = c₁ q(E₁) + c₂ q(E₂) + c₃ q(E₃)
     = c₁ (3, 4, 3) + c₂ (1, 1, 1) + c₃ (48, 24, 11)
     = (3 c₁ + c₂ + 48 c₃, 4 c₁ + c₂ + 24 c₃, 3 c₁ + c₂ + 11 c₃).

Substituting for X in terms of its components, c_i's, results in the following equation:

q(c₁, c₂, c₃) = (3 c₁ + c₂ + 48 c₃, 4 c₁ + c₂ + 24 c₃, 3 c₁ + c₂ + 11 c₃).

The RHS is a vector Y in the LC vector space, hence, a linear combination of the basis vectors: d₁ A + d₂ L + d₃ M = (d₁, d₂, d₃). Its substitution in the above equation gives:

(3 c₁ + c₂ + 48 c₃, 4 c₁ + c₂ + 24 c₃, 3 c₁ + c₂ + 11 c₃) = (d₁, d₂, d₃).

In terms of matrices the equation takes the form QX = Y, where

$Q\; =$ $\left(\begin{array}{ccc}3& 1& 48\\ 4& 1& 24\\ 3& 1& 11\end{array}\right)$,

$X\; =$ $\left(\begin{array}{c}\mathrm{c₁}\\ \mathrm{c₂}\\ \mathrm{c₃}\end{array}\right)$,

$Y\; =$ $\left(\begin{array}{c}\mathrm{d₁}\\ \mathrm{d₂}\\ \mathrm{d₃}\end{array}\right)$.

Q is the matrix corresponding to the linear map q that takes a (co)bif vector X and returns its LC vector Y. In particular, the basis vectors E_i's (standing for verses 2:0, 2:1 and 2:286) are mapped to the internal LCs of those verses. The basis vectors in standard form (1, 0, 0), (0, 1, 0) and (0, 0, 1) are thus mapped under matrix multiplication by Q to the linear contents of their corresponding verses.

The multiplications of LC vectors so far are obtained indirectly via the linear map q where the vectors are pulled back into the (co)bif algebra where their products are computed, then mapped back into LC via q. To sever this dependence to allow computing the products directly in LC requires defining the multiplication table among its basis vectors A, L, and M. The multiplication will then extends linearly to all vectors in LC. This is achieved by pulling the basis letters back the same way, compute their pairwise products and map them back. With F_i's standing for the basis letters, let X_i's be their pullback under q: X_i = q⁻¹(F_i), for 1 ≤ i ≤ 3.

The inverse of Q is:

$Q-1=$ $\left(\begin{array}{ccc}\mathrm{-1373}& 1& 1372\\ \mathrm{-5833}& \mathrm{-3}& 5837\\ 2059& 0& \mathrm{-2059}\end{array}\right)$.

It takes an LC and returns its versor in the (co)bif-2 algebra. The basis LCs F_i represented in standard form (1, 0, 0), (0, 1, 0) and (0, 0, 1) transform under Q⁻¹:

X₁ = Q⁻¹F₁ = Q⁻¹ (1, 0, 0) = (−1373, −5833, 2059)

X₂ = Q⁻¹F₂ = Q⁻¹ (0, 1, 0) = (1, − 3, 0)

X₃ = Q⁻¹F₃ = Q⁻¹ (0, 0, 1) = (1372, 5837, −2059)

The pairwise products X_iX_j for 1 ≤ i,j ≤ 3 in the (co)bif-2 algebra are:

(L):

X₁X₁ = (-807, -933, -5466)
X₁X₂ = (-4268, -6075, 1406)
X₁X₃ = (-2247, 4234, 3767)

X₂X₁ = (3821, -4746, -4073)
X₂X₂ = (-3382, 781, -640)
X₂X₃ = (5742, -6016, 317)

X₃X₁ = (1360, -4102, 3413)
X₃X₂ = (1752, 2027, 3614)
X₃X₃ = (-3894, 1742, 4617).

Under Q the products are mapped back to LC vector space:

(L):

AA  = Q(X₁)Q(X₁) = Q(X₁X₁) = (915, 4322, 5)
AL  = Q(X₁)Q(X₂) = Q(X₁X₂) = (-2179, -2100, -3413)
AM = Q(X₁)Q(X₃) = Q(X₁X₃) = (551, -3225, 839)

LA  = Q(X₂)Q(X₁) = Q(X₂X₁) = (1668, 1665, 5)
LL  = Q(X₂)Q(X₂) = Q(X₂X₂) = (-1994, -2713, -3708)
LM = Q(X₂)Q(X₃) = Q(X₂X₃) = (1032, -834, 2000)

MA  = Q(X₃)Q(X₁) = Q(X₃X₁) = (-1259, -5629, -570)
ML  = Q(X₃)Q(X₂) = Q(X₃X₂) = (2997, -5805, -3751)
MM = Q(X₃)Q(X₃) = Q(X₃X₃) = (-4173, -4602, 2756)

which then constitute the multiplication table for the left LC algebra.

Similar construction leads to the following multiplication table for the right LC algebra:

(R):

X₁X₁ = (771, 657, 2045)
X₁X₂ = (-5580, 5812, 3066)
X₁X₃ = (1290, -5290, 1556)

X₂X₁ = (4360, 3233, -1599)
X₂X₂ = (-6062, 5590, 3464)
X₂X₃ = (3942, 3195, -4075)

X₃X₁ = (-5406, 5872, 2627)
X₃X₂ = (-2782, 5851, -4637)
X₃X₃ = (-156, -4106, 1249).

and their corresponding mapped products to LC vector space under Q:

(R):

AA  = (-446, 2033, 71)
AL  = (-3427, 6288, -2596)
AM = (-2914, -877, 2999)

LA  = (3046, -5006, -1276)
LL  = (1312, 993, 114)
LM = (-2821, -2655, -4410)

MA  = (1477, -3492, 5854)
ML  = (3475, -2292, -2714)
MM = (4590, -148, -3532)

The pair of left/right ALM algebras are isomorphic to their corresponding left/right (co)bif-2 algebras. The multiplications among versors have thus been transferred to the initial letters A, L, and M. The LC algebra will serve as a foundation for further exploration of specific verses in chp 2.