The set of all verses forms a magma algebra over the ring of integers ℤ as was described previously. With explicit presence of verses, the algebra is fully determined when the pairwise products of all verses are known. Since, the multiplication is noncommutative (UV ≠ VU, for arbitrary verses U and V) there are about 4 × 107 pairs to be evaluated. It is therefore advantageous to have the algebra over a field (a vector space with a multiplication operation). The choice of a small suitable basis would then allow the verses to be expressed in the terms of the basis vectors, resulting in a significant simplification. To illustrate the approach, let B = {E1, E2, E3, E4} be a basis consisting of four arbitrary verses denoted by Ei's appearing in their natural ascending order: Ei < Ej for i < j. The choice of the verses in the basis is not the prime focus at this stage. The verses Ei's will be made explicit when the vector space is eventually implemented. With the basis fixed, all verses will be represented in this four-dimensional vector space as unique linear combinations of the four basis vectors (also commonly referred to simply as their coordinate vectors). The verses and their coordinate vectors (relative to the ordered basis B) are considered equivalent and will be treated interchangeably.
Finding the coordinate representations of the basis vectors is
straightforward since they are linearly independent (one can not be
non-trivially expressed in terms of the others). A basis vector expressed
in terms of itself and other basis vectors necessarily uses a coefficient
of 1 for itself and 0s for all other basis vectors:
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 vectors on the RHS with 1 in the i-th position and zeros
elsewhere form the standard ordered basis representing Ei.
All other verses similarly have unique coordinate representations. Let V be a verse not in the basis. Its coordinate vector can be written as (c1, c2, c3, c4) with components ci's in some appropriate field to be specified. The components ci's will be determined by setting up links between the verse V and the basis verses Ei via directed intervals. The linking is subsequently quantified via directed lengths that establish a measure of interaction between the dependent verse V and the basis.
Given a pair of verses U and V, the directed interval DI = [U, V] is the
ordered set consisting of all consecutive adjacent verses starting at U
and ending at V. It is positively oriented when U < V (ascending
sequence) and negatively oriented if U > V (descending sequence):
[U, V] = {U, U + 1, U + 2, ⋅ ⋅ ⋅ , V}
for U < V
[U, V] = {U, U − 1, U − 2, ⋅ ⋅ ⋅ , V}
for U > V
U + 1 is the adjacent verse immediately after U and likewise U − 1
is the adjacent verse immediately before U. The verses U and V are the
two endpoints of a sequence of verses starting at the first endpoint and
continuing with all the intervening adjacent verses and ending at the
second endpoint. The directed or signed length of the directed interval,
denoted by ∥ ⋅ ∥, is the total number of verses in
the sequence with the orientation taken into consideration. It is positive
for positively oriented interval and negative if it is oriented
negatively. For instance, [1:5, 2:3] = {1:5, 1:6, 1:7, 2:0, 2:1, 2:2, 2:3}
is an ascending sequence of seven consecutive verses starting at 1:5 and
ending at 2:3. It is positively oriented (1:5 < 2:3) with
∥[1:5, 2:3]∥ = 7. In contrast [2:3, 1:5] =
{2:3, 2:2, 2:1, 2:0, 1:7, 1:6, 1:5} is descending (2:3 > 1:5)
and hence, it is negatively oriented: ∥[2:3, 1:5]∥ =
−7. In the exceptional case of a directed interval starting and
ending at the same verse, a singleton, [U, U] = {U}, the orientation is
taken to be positive: ∥[U, U]∥ = 1.
Given a directed interval [U, V] and a verse W, their product is defined as [U, V]W = [UW, VW]. The new directed interval is similarly specified by its two endpoint verses UW and VW. The two products are carried out in the magma. It is a sequence of verses starting at UW and ending at VW with all the intervening adjacent verses. The new directed interval has its own independent orientation regardless of the original interval. It is entirely determined by the new endpoints: positive for UW < VW and negative for UW > VW. The product interval can expand, shrink, or remain the same in length. Since the magma is noncommutative, switching the order of the product gives another distinct interval: W[U, V] = [WU, WV] with its own independent length and orientation.
The product of a directed interval and a verse can be extended
to formal sums of verses. Given an interval [U, V] and verses
W1 and W2 the followings hold:
[U, V](c1W1 + c2W2) =
c1[U, V]W1 +
c2[U, V]W2
=
c1[UW1, VW1] +
c2[UW2, VW2]
and
(c1W1 + c2W2)[U, V] =
c1W1[U, V] +
c2W2[U, V]
=
c1[W1U, W1V] +
c2[W2U, W2V]
The coefficients c1 and c2 are from the field.
The directed length ∥ ⋅ ∥ can similarly be
extended to formal sums of directed intervals:
∥c1[U1, V1] +
c2[U2, V2]∥ =
c1∥[U1, V1]∥ +
c2∥[U2, V2]∥
In particular, the two directed intervals [U, V] and [V, U]
are oppositely oriented and their directed lengths are equal in
magnitude but opposite in sign:
∥[U, V]∥
= −∥[V, U]∥
or
∥[U, V] + [V, U]∥ = ∥[U, V]∥ +
∥[V, U]∥ = 0.
Note that a single directed
interval can not be of zero length: ∥[U, V]∥ ≠ 0.
With the directed intervals and their lengths, the links between the
basis verses and the dependent verse V are made explicit making
it possible to determine the coordinate vector of V,
(c1, c2, c3, c4) with
components ci's in an appropriate field to be specified.
For each basis vector Ei (1 ≤ i ≤ 4), let
Di = [Ei, V] be the directed interval it forms
with V. Fully expanded in terms of the basis vectors V takes the
following form:
V = ∑
cj Ej, for 1 ≤ j ≤ 4, (Eq. *)
Turning the equation around and right-multiplying by
Di on both sides gives:
(
∑
cj Ej
)
Di = VDi,
(1 ≤ i,j ≤ 4)
The LHS simplifies to
∑ cj
EjDi = VDi,
(1 ≤ i,j ≤ 4)
There are four equations that are indexed
by i (one for each Di), while the components within each
equation are indexed by j. Substituting for Di in terms
of its interval form gives:
∑ cj
Ej[Ei, V] =
V[Ei, V], (1 ≤ i,j ≤ 4)
Carrying out the products gives:
∑ cj
[EjEi, EjV] =
[VEi, VV], (1 ≤ i,j ≤ 4)
Applying the directed length ∥ ⋅ ∥
to both sides gives:
∥
(
∑ cj
[EjEi, EjV]
)
∥ = ∥ [VEi, VV] ∥,
(1 ≤ i,j ≤ 4)
Which reduces to the following:
∑ ∥
[EjEi, EjV]
∥ cj = ∥ [VEi, VV] ∥
This is a linear system (LSYS) of four equations, with
cj's as the four unknowns. The directed intervals and their
lengths are all known quantities. The solutions to the LSYS equations
will determine the coordinate components of V relative to the
basis B. The coordinate vectors are the linear
representations of verses in the vector space and they will be
referred to as the versors. Since, the deriving equations were
obtained via the right-multiplication of the directed interval,
the result is the right versor of V.
Symmetrically, the left versor is obtained in exactly the same way
but via the left-multiplication by
Di on both sides of (Eq. *):
Di
(
∑
cj Ej
)
= DiV,
(1 ≤ i,j ≤ 4)
Which likewise simplifies to
∑ cj
DiEj = DiV,
(1 ≤ i,j ≤ 4)
Replacing the Di once again
with its interval form and similarly carrying out the products inside
the sum followed by applying the directed length on both sides gives:
∑
∥ [EiEj, VEj] ∥
cj = ∥ [EiV, VV] ∥
The solutions to the subsequent LSYS of four equations
similarly determine the four unknowns, cj's. The
resulting coordinate vector is the corresponding left versor of V.
In the original magma algebra the verses directly participated in the algebra necessitating evaluations of a large number of pairwise products. By contrast, when verses are represented by their versors (their linear representations in the vector space) the algebra can be completely specified with the pairwise products of only the basis vectors since the multiplication will then extends linearly to all vectors in the vector space, thus reducing the complexity of the setup.