2:282 ALGEBRAS AND POLY-LCs

The actions of the left/right stabilizers of 2:282 ILC (U) on its respective left/right XLC (V) were abstracted into corresponding left/right unital polynomial algebras of noncommuting indeterminates with nonassociative products. It is interesting to consider what algebra is obtained if the actions of the same stabilizers are applied to U itself. By definition, the stabilizers leave the ILC intact: S_i U = U, for 1 ≤ i ≤ 4. Hence, the stabilizers act trivially like the identity operator: S_i = Id. Their abstractions would lead to a polynomial algebra whose indeterminates are constant: X_i = 1. The algebra of constant polynomials over the finite field F12697 is simply the ground field itself.

The actions when restricted to the XLCs resulted in a pair of polynomial algebras. However, the list of algebras can be vastly expanded by considering the interactions between U and V. Both are vectors in the LC-algebras and as such, their products UV, VU and their higher degree (bracketed) monomials are all vectors and, likewise, can be acted on by the stabilizers. Each actions on such products would lead to polynomial algebras as before.

Monomials can be thought of as words formed with U and V serving as letters, also know as bitstrings. The alphabet over which the words can be formed consists of {U, V} with the chosen ordering U < V. The words can subsequently be placed in lexicographical order, starting with 1-words U and V, followed by 2-words UU and UV, ... , then 3-words, and so forth: U, V, UU, UV, VU, VV, UUU, ... , VVV, UUUU ... , VVVV, etc. Such an ordered list of words can be placed side-by-side with the list of positive numbers indicating their positions within the list :

1   U
2   V
3   UU
4   UV
5   VU
6   VV
7   UUU
8   UUV
9   UVU
.   .
.   .
.   .

The correspondence between the two columns, numbers and words, can be explicitly established if the bitstrings are treated as binary numbers with U and V standing for 1 and 2, resp., e.g., the bitstring VUUVVVU corresponds to 2112221, or 205 in decimal:

VUUVVVU = 2112221₂ = 2 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 2 × 2³ + 2 × 2² + 2 × 2¹ + 1 × 2⁰ = 205,

which indicates that the word VUUVVVU is at position 205 on the list. This is the bijective base two (BB2) representation of numbers with 1 and 2 as digits (bits). This is in contrast with the usual base two (binary) number system with digits 0 and 1. It is bijective in the sense that the representations of numbers are unique. In the conventional base B number system, where the digits go from 0 to B − 1, the leading zeros spoil uniqueness: e.g., 15, 015, 0015, ... , all represent the same number in decimal system. In bijective systems, however, where digits run from 1 to B this can not happen as there is no zero digit.

Bitstrings in U and V viewed as BB2 numbers allow defining actions of the stabilizers on arbitrary positive numbers. Given a positive number it is first expressed in BB2, followed by U and V (ILC and right XLC) replacing digits 1 and 2, resp. The resulting bitstring is a monomial which can then be multiplied (with appropriate bracketings) in the right LC-algebra, resulting in vectors which can subsequently be acted on by the right stabilizers S_i's. Symmetrically, the actions can be extended to the negative numbers where U is alternatively paired with the left XLC V. The resulting (bracketed) monomial is then evaluated in the left LC-algebra with subsequent actions by the left stabilizers. Here, left and right have been associated respectively with negative and positive numbers similar to the number-axis with negatives on the left and positives on the right, with zero in the middle as the origin. Viewed in this light, the actions on negatives can be formally stated as follows. For N > 0, then −N < 0. Let S_l and S_r be a pair of matching left and right stabilizers. The action on the negative number, −N, is defined:

S_r (−N) = (−S_r) N = S_l N,

where S_l is treated as the "negative" of S_r. In other words, the action of the right stabilizer on a negative number is defined to be the action of the matching left stabilizer on the corresponding positive number.

The domain of actions can be further extended to include zero. Since zero is not represented by any BB2 bitstring in U and V, the zero-vector $\overrightarrow{0}$ can be formally taken as the appropriate representation of number 0. Then the action on 0, (S 0), is defined as S $\overrightarrow{0}$ = $\overrightarrow{0}$ for any linear operator S, resulting in a singleton consisting of the zero-vector {$\overrightarrow{0}$}. The only algebra possible on a singleton is the trivial algebra where all sums and products are zero. With this setup the actions are thus defined on the set of all integers. The two left and right polynomial algebras derived previously (actions on left and right V) can be seen as actions on −2 and +2, resp. The left and right stabilizers actions on the ILC, U, resulted in the ground field, itself. This is the action on both −1 and +1.

With 1-word bitstrings, (U and V, left/right actions on ±1 and ±2), there is no interplay between the LC-pair. However, longer words allow for interactions between the internal and external LCs. The 2-words UU, UV, VU, VV correspond to BB2 expressions 11, 12, 21, 22, resp., or in decimal, to 3, 4, 5, 6. The left and right actions on the four words constitute the actions on {−6, −5, −4, −3, +3, +4, +5, +6} resulting in eight polynomial algebras. The longer k-words for k ≥ 3, (corresponding to integers ≤ −7 or ≥ +7), require bracketing as the products in the LC-algebras are nonassociative. There are eight 3-words from UUU to VVV, (positions 7 through 14 on the list), each with two bracketings: (⋅⋅)⋅ and ⋅(⋅⋅) for a total of 16 nonassociative words. Similarly, the 16 4-words coupled with 5 bracketings each give 80 bracketed 4-words. In general, there are 2^k k-words on the 2-letter alphabet {U, V} with Catalan, C_k, bracketings per word for a total of 2^k C_k parenthesized words. However, not all such words would result in distinct vectors when they are evaluated in the LC-algebras. The algebras are over the finite prime field F12697 with 12697 elements: from 0 to 12696. An arbitrary vector (LC) is of the form (a, l, m), where the components are the counts of A, L, M letters. The total number of vectors, therefore, is 12697³ ≈ 2 × 10¹³. On the other hand, the list of words is infinite. Since each (bracketed) word evaluates to a vector, infinitely many words evaluate to a single vector. This indicates that distinct numbers (represented as BB2 bitstrings) can lead to the same vector and thus are identical under the actions of the stabilizers. When distinct numbers, represented as BB2 bitstrings in U and V, lead to the same vector they are considered equivalent. This equivalence is a form of loosening and weakening of equality. Equality is strict and rigid: a number can only be equal to itself and nothing else. An equivalence relation relaxes this strict requirement and treats distinct numbers "the same" for a specific purpose. Under this broader notion of equivalence, the set of numbers are partitioned into disjoint subsets, the equivalence classes, where each such class contains equivalent numbers. There are as many partitions as there are vectors in the LC-algebras.

The algebras derived from the U and V bitstrings account for the interactions between the internal and external LCs of 2:282. Purely external algebras can also be obtained solely based on the interactions between left and right XLCs. Let V and W be the left and right XLCs, resp. A mutual pair of polynomial algebras can be derived where stabilizers of V act on W and symmetrically the stabilizers of W act on V. This can be extended to bitstrings on the {V, W} alphabet representing BB2 bits {1, 2}. The complement of a bitstring is another bitstring where V and W are swapped: V ⟺ W, e.g. VVWVWWWWVVWV and WWVWVVVVWWVW form a complement pair. In a complement-pair of bitstrings, the stabilizers of one can act on the other giving a complementary pair of algebras. The complementation can also be applied to an arbitrary positive integer where it is first expressed in BB2 bitstring, then complemented. For example decimal 1000 in BB2 is 222212112 (WWWWVWVVW) with complement 111121221 (VVVVWVWWV) corresponding to the decimal 533. The pair 1000 and 533 would then give rise to a complementary pair of algebras.

The stabilizers S_i's preserved the ALM ILC in 2:282. The ILC, U = (107, 66, 31) accounted for all occurrences of the ALM letters in the verse. In general, ILC can be a proper subset of such letters. For instance, it can be the counts of letters only at specific locations such as at even or odd positions. It can be the counts of letters that appear in specific sentences. It can be the letters that appear in a selected set of words in accordance with some criteria, etc. The vectors in the algebra derived from these letters are the linear contents (LCs) whose components carry information such as the counts of such letters, etc. The LCs can be constructed in essentially two distinct ways. In the direct method the LC is of additive type (AdLC) and simply keeps track of the counts of distinct letters in the set. No other information is retained. The indirect method of multiplicative type (MuLC) incorporates additional information regarding the positions and ordering of the letters as well as word boundaries and sentence structures. This can be achieved if the sequence of the chosen subset of letters, taken exactly as they appear in the text, form a monomial (a single-term expression) which can then be evaluated (multiplied) in the algebra. Since the LC-algebra is noncommutative the ordering of letters are thus accounted for. Nonassociativity allows inclusions of additional information regarding textual structures. A word-boundary-respecting bracketing carries information regarding the break up of text into words. In general, text splittings into sentences, phrases, words and other sub-structures of interest can be reflected in appropriately chosen bracketings. The finial bracketed expression, when multiplied out, will result in vector(s) which can serve as the internal LCs (ILCs). The AdLC together with the MuLC(s) provide a more complete account of the internal properties of verses. The subsequent algebraic properties of the LCs are reflections of the counts of the letters as well as their positions and groupings within the text.

As an example of a proper subset of letters in 2:282 is the ILC formed with the ALMs entirely within the first sentence. It consists of ten words all with the exception of the sixth contain a total of 18 ALM letters. The AdLC is simply the counts of the distinct letters: AdLC = 11 A + 3 L + 4 M = (11, 3, 4). The derivation of the MuLC begins with the sequence of the ALM letters as they appear in the sentence: A-A-A-L-A-M-A-A-A-A-M-A-L-A-L-M-M-A. The dashes "-" represent the remaining intervening non-ALM letters in the sentence. Removing the dashes will result in an ALM monomial of length 18, AAALAAMAAAAMALALMMA, to be evaluated in the LC algebras. However, parenthesization is necessary due to nonassociativity. The choice of bracketing is based on the division of the monomial into subwords as inherited from the word structure of the sentence:

"AA AL AMA AA AM AL AL MM A",

where the ALM letters within the same word in the original sentence are grouped together. The absence of ALM letters in the sixth word in the sentence implies that the first five and the last four ALM-words are each consecutive subsequences that are separated by the non-ALM-word. This immediately suggests at the outset the folowing partial parenthesization into two encompassing factors:

(AA AL AMA AA AM) (AL AL MM A).

Word-respecting parenthesization further isolates each ALM-word:

( (AA)(AL)(AMA)(AA)(AM) ) ( (AL)(AL)(MM)A ).

The first three parenthesized-words (AA)(AL)(AMA) form the phrase "O you who believe" which makes frequent appearances in many verses. Their bracketing into one grouping signifies this fact:

( ( (AA)(AL)(AMA) ) (AA)(AM) ) ( (AL)(AL)(MM)A ).

The two remaining forth and fifth words (AA)(AM) can similarly be grouped together as they relate to financial matters/loan transactions:

( ( (AA)(AL)(AMA) ) ( (AA)(AM) ) ) ( (AL)(AL)(MM)A ).

The third word in the first factor, (AMA), splits further into AM and A, as the first two adjacent letters are separated from the last A, implying parenthesization (AM)A, displayed as AM.A:

( ( (AA)(AL)(AM.A) ) ( (AA)(AM) ) ) ( (AL)(AL)(MM)A ).

In the second factor, the first three words (AL)(AL)(MM) form the phrase "until a specific time" and are thus bracketed together:

( ( (AA)(AL)(AM.A) ) ( (AA)(AM) ) ) ( ( (AL)(AL)(MM) )A ).

Furthermore, in the same phrase, the first (AL), "until", a proposition, can be separated from (AL)(MM), "specific time":

( ( (AA)(AL)(AM.A) ) ( (AA)(AM) ) ) ( ( (AL) ( (AL)(MM) ) )A ).

This is the parenthesization to the extent that is implied by the structure of the sentence based on words, subwords, propositions and phrases. However, it is not, thus far, a fully bracketed expression ready to be evaluated. The three words in the first phrase, (AA)(AL)(AM.A), can further be bracketed in two ways: ( (AA)(AL) ) (AM.A) and (AA) ( (AL)(AM.A) ). Altogether, the two fully parenthesized monomials can be multiplied resulting in two MuLCs that can be evaluated in the left/right LC algebras.

The restriction of the ILC to the first sentence calls for similar restriction on the left/right LC algebras. Recall that the LC algebras were obtained from the (co)bif algebras via the bijective map Q transporting the algebra structure and turning Q into algebra isomorphism. The versors thus acquired LCs via Q whose columns were the full ILCs of the basis verses {2:0, 2:1, 2:286}:

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

The first two verses 2:0 and 2:1 are already one sentence long. The third basis verse 2:286 has the total ILC (48, 24, 11). When it is restricted to the first sentence the new ILC is (11, 7, 2), resulting in:

$Q\; =$ $\left(\begin{array}{ccc}3& 1& 11\\ 4& 1& 7\\ 3& 1& 2\end{array}\right)$

The resulting left/right S1 LC algebras (S1: restricted to sentence one) have the following multiplication tables:

(L):

AA  = (124, 4328, 1280)
AL  = (8481, 8561, 11038)
AM = (10163, 4425, 2561)

LA  = (4897, 3399, 9827)
LL  = (8989, 8167, 2052)
LM = (9805, 10496, 588)

MA  = (1159, 7362, 9827)
ML  = (9749, 12467, 103)
MM = (4281, 6791, 8565)

(R):

AA  = (8502, 2956, 909)
AL  = (2573, 6608, 9443)
AM = (4414, 5521, 9511)

LA  = (11856, 73, 12409)
LL  = (114, 5590, 7029)
LM = (7852, 12542, 7577)

MA  = (6986, 9106, 4281)
ML  = (4814, 11398, 12083)
MM = (10884, 5188, 12277)

The two multiplicative LCs, MuLC₁ and MuLC₂, can now be evaluated in the left/right S1 LC algebras:

MuLC₁ = ( ( ( (AA)(AL) ) (AM.A) ) ( (AA)(AM) ) ) ( ( (AL) ( (AL)(MM) ) )A )

      = (3464, 3979, 10665)       (Left)

      = (6598, 2307, 124)       (Right)

MuLC₂ = ( ( (AA) ( (AL)(AM.A) ) ) ( (AA)(AM) ) ) ( ( (AL) ( (AL)(MM) ) )A )

      = (3579, 3790, 7753)       (Left)

      = (11169, 11530, 9133)     (Right)

Consequently, the MuLCs acquire handedness as they are evaluated in the left/right S1 LC algebras.

The AdLC and MuLCs are constructed in entirely different ways and represent complementary aspects of the first sentence. The AdLC accounts for the counts of the ALM letters while MuLCs account for their positions and groupings into words and phrases. Thus far, they remain separate and disconnected. Although, they can be treated independently, it is advantageous if the three ILCs, {AdLC, MuLC₁, MuLC₂}, are consolidated by combining them into a unified whole, the total ILC. In one approach, the total ILC is simply their sum:

ILC = AdLC + MuLC₁ + MuLC₂.

Here, the separate contributions from the three LCs are merely lumped together in a sum and lose their original distinctive identities. A more refined approach would have the three LCs retain their individual uniqueness and characteristics. This can be achieved if the two MuLCs are distinguished with symbols X and X², forming a quadratic polynomial in X:

ILC = AdLC + MuLC₁ X + MuLC₂ X².

Each LC component in the polynomial is now identified with a distinct power of X. The AdLC is associated with X⁰ = 1. Substituting for the three LCs with their derived values give:

ILC = (11, 3, 4) + (3464, 3979, 10665) X + (3579, 3790, 7753) X²     (Left)
ILC = (11, 3, 4) + (6598, 2307, 124) X + (11169, 11530, 9133) X²     (Right)

The right-hand-sides can be further simplified via the usual addition and scalar multiplication of vectors:

ILC = (3579 X² + 3464 X + 11, 3790 X² + 3979 X + 3, 7753 X² + 10665 X + 4)     (Left)
ILC = (11169 X² + 6598 X + 11, 11530 X² + 2307 X + 3, 9133 X² + 124 X + 4)     (Right)

In fully expanded form they are:

ILC = (3579 X² + 3464 X + 11) A + (3790 X² + 3979 X + 3) L + (7753 X² + 10665 X + 4) M     (Left)
ILC = (11169 X² + 6598 X + 11) A + (11530 X² + 2307 X + 3) L + (9133 X² + 124 X + 4) M     (Right)

The coefficients of A, L, M that formerly were merely numbers from the finite field F12697 are now enriched and elevated into polynomials in X over the same field, denoted by F₁₂₆₉₇[X]. Note that the two expressions for the ILCs reduce to the same original (number) form (with no handedness) when X = 0:

ILC = 11 A + 3 L + 4 M = (11, 3, 4),

where the reduced ILC accounts only for the counts of letters. The original (purely number-based) ILCs can be seen as special cases where vectors have constant components which are simply zero-degree polynomials. The presence of polynomials signify the inclusion of the first-sentence internal structures in addition to its counts of ALM letters.

Replacing the scalars from the finite field to polynomials over the same field makes the algebraic framework significantly more expressive. It transforms the static coefficients into dynamic variables capable of encoding more information. It augments the purely algebraic setting with geometric and analytic flavor. Accordingly, the proper handling and treatment of the new polynomial-based ILCs and the applicable operations on them demand a corresponding need for appropriately constructed polynomial based LC-algebras over F12697.