(CO)BIF ALGEBRAS

The linear representations of verses (left and right versors) were obtained through the applications of the left/right actions of the directed interval DI operator relative to a fixed unspecified ordered basis. It led to a set of linear systems of equations whose solutions were the vector representations of the verses in the vector space. To establish the vector space the choice of the basis will be made explicit. A small subset of verses will be selected to serve as the basis. It consists of four verses, the first and last verses in chps one and nine, respectively: B = {1:1, 1:7, 9:1, 9:127}, also denoted as before by {E₁, E₂, E₃, E₄}, with E_i's standing for the basis verses in their natural ascending order. All remaining verses will be represented in the vector space as unique linear combinations of the four basis vectors. Since, the basis consists of verses from chapters 1 and 9, the resulting algebras will be labeled as (co)bif-19 algebras.

Given a verse V not in the basis, let its versor be (c₁, c₂, c₃, c₄) with components c_i's in some appropriate field to be specified. The components are the solutions of the matrix equation AX = B, with A of dim 4 × 4 and column vectors X and B each of dim 4:

A = (a_ij) for 1 ≤ i,j ≤ 4

X = (c_i) for 1 ≤ i ≤ 4

B = (b_i) for 1 ≤ i ≤ 4

where

a_ij = ∥ [E_jE_i, E_jV] ∥ for 1 ≤ i,j ≤ 4

b_i = ∥ [VE_i, VV] ∥ for 1 ≤ i ≤ 4

The known quantities (the entries in A and B) are the directed interval lengths. They are strictly bounded by the total number of verses: −6346 < ∥ ⋅ ∥ < +6346. The choice of the finite field for the scalars is therefore entirely based on the sole requirement that it must be sufficiently large to hold all directed lengths from −6345 to +6345. Hence, the field must contain at the minimum 12691 (2 × 6345 + 1) elements. The size of a finite field, (its number of elements), is a power of a prime. For simplicity a prime field will be chosen. Since, 12691 is composite, the prime immediately after, 12697, will be chosen. This will establish the prime field F12697 as the proper field of scalars for the vector space. Absent other requirements, this ensures that the ground field is not unnecessarily large.

Given two arbitrary verses U and V, with their respective versors (c₁, c₂, c₃, c₄) and (d₁, d₂, d₃, d₄), they can be expanded in terms of the basis vectors:

U = ∑ c_i E_i,   for 1 ≤ i ≤ 4
V = ∑ d_j E_j,   for 1 ≤ j ≤ 4,

with coefficients c_i, d_j ∈ F12697. It follows that their product,

UV = ∑ ∑ c_id_j E_iE_j,   for 1 ≤ i,j ≤ 4,

is fully determined in terms of the products of the basis vectors, E_iE_j. The pairwise products of the basis vectors are carried out in the the magma as was previously shown. The multiplications would then extend linearly to all vectors in the vector space. In the magma, the product of any two verses is another verse. In particular, the product of a pair of basis verses is simply another verse, not necessarily a basis verse. Therefore, it, too, can be expressed as a linear combination of the basis vectors:

E_iE_j = ∑ c_ijk E_k,   for 1 ≤ i,j,k ≤ 4,

where, the product verse E_iE_j is expressed as a linear combination of the basis vectors E_k. The coefficients c_ijk are the structure constants of the algebra. They act as the multiplication table with respect to basis B. There are a total of 64 (4 × 4 × 4, for 1 ≤ i,j,k ≤ 4) structure constants. Let the 16 verses V_ij = E_iE_j (for 1 ≤ i,j ≤ 4) be the the pairwise products of the basis vectors E_i and E_j obtained in the magma. Their versors (c_ijk, for 1 ≤ k ≤ 4) fully determine the multiplication table:

E_iE_j = V_ij = (c_ijk) = (c_ij1, c_ij2, c_ij3, c_ij4)   for 1 ≤ i,j ≤ 4.

Since the product verses V_ij are represented by their pairs of left and right versors, each handedness will lead to a distinct multiplication table and hence a separate (co)bif-19 algebra: the left versors define the left algebra and symmetrically, the right versors generate the right algebra. The followings are the two multiplication tables for the left and right (co)bif-19 algebras over F12697.

(Left):

E₁E₁ = (1 0 0 0) = E₁
E₁E₂ = (-2909, -3377, 5372, -5867)
E₁E₃ = (673, -5777, 2006, 300)
E₁E₄ = (-2893, -4362, -1974, 4319)

E₂E₁ = (1 0 0 0) = E₁
E₂E₂ = (1341, 2575, 131, 5952)
E₂E₃ = (1796, -335, -2541, -173)
E₂E₄ = (-5405, 3093, 2124, -5590)

E₃E₁ = (1 0 0 0) = E₁
E₃E₂ = (-3341, -1141, 2923, -765)
E₃E₃ = (-6092, -3529, -3238, -5444)
E₃E₄ = (-415, -3344, -4184, -1792)

E₄E₁ = (1 0 0 0) = E₁
E₄E₂ = (4344, 1920, -5301, 5564)
E₄E₃ = (4792, -5223, 5553, 3634)
E₄E₄ = (375, 1584, -4703, 5204)

(Right):

E₁E₁ = (1 0 0 0) = E₁
E₁E₂ = (11, 1401, -1200, -3285)
E₁E₃ = (-2927, -3876, 1302, 67)
E₁E₄ = (2424, 2097, -5774, -4076)

E₂E₁ = (1 0 0 0) = E₁
E₂E₂ = (2992, -1736, 386, 3487)
E₂E₃ = (-4871, -6169, 2173, 5930)
E₂E₄ = (137, 5392, 3550, 3967)

E₃E₁ = (1 0 0 0) = E₁
E₃E₂ = (-2476, 5182, 1922, 2855)
E₃E₃ = (4265, -100, -3609, -2824)
E₃E₄ = (2498, 3917, -475, 2885)

E₄E₁ = (1 0 0 0) = E₁
E₄E₂ = (-3186, -2455, -6322, 3467)
E₄E₃ = (-2489, -5021, 2158, 926)
E₄E₄ = (1068, -5651, 5450, -5392)

The products E_iE_j are expressed on the RHS as vectors whose components are the structure constants c_ijk. They can also be expanded explicitly in terms of the basis vectors. For instance, the last equation will take the following form when fully expanded:

E₄E₄ = 1068 E₁ − 5651 E₂ + 5450 E₃ − 5392 E₄.

Note that E_iE₁ = E₁ for 1 ≤ i ≤ 4, in both algebras as E₁ is the right-absorbing element in the magma as was previously described. This immediately leads to zero-divisors in the algebras: let V = E₂ − E₃. Then:

VE₁ = (E₂ − E₃)E₁ = E₂E₁ − E₃E₁ = E₁ − E₁ = 0.

The (co)bif-19 algebras are noncommutative and nonassociative as they are derived from the magma. They are of dimension 4 (four basis verses in B) defined via structure constants. The algebras are fully specified with only 16 (4²) pairwise products of the basis vectors. This is in sharp contrast to the magma algebra which requires a much larger set of (6346²) pairwise products of all verses. The magma is to serve as the foundation for the (co)bif algebras of varying dimensions adapted for more specific purposes.