The total ILC of the first sentence in 2:282 were derived in the
left/right S1 LC algebras and took the form
ILC = a(X) A + l(X) L +
m(X) M = (a(X), l(X), m(X))
with components a(X), l(X) and m(X) as
quadratic polynomials in F12697[X], the set of polynomials in
single indeterminate X with coefficients in the finite field F12697.
Recall, F12697 has the underlying set of 12697 elements: {0, 1, 2, 3, 4,
... , 12696}. As the new LCs are polynomial-based, the proper setting for
their algebraic manipulations calls for similar enhancement of the
original left/right LC algebras.
The original LC algebras were based purely on numbers. The underlying vector space was based on F12697; the scalars and the vector components were numbers from the finite field. An enhanced version will be developed based on F12697[X]. The polynomials are of the general form c0 + c1 X + c2 X2 + c3 X3 + ...., with coefficients ci's in F12697. Although, the coefficients are bounded (0 ≤ ci ≤ 12696) the set of such polynomials is still infinite since the degrees can grow without limits. If R(X) and S(X) are polynomials of degrees m and n, resp, with positive integers m and n, then their product R(X)P(X) is of degree m + n. The degree of the sum (R(X) + S(X)) however can decrease as there may be cancellations of similar terms.
The transport of the algebra structures from (co)bif algebras to the LC vector spaces were carried out via the bijective map Q (ultimately an algebra isomorphism by construction) whose columns were the ILCs of the basis vectors {2:0, 2:1, 2:286}. The derivation of the revised Q will proceed along similar lines. The ILCs are first restricted to the first sentences. The internal structures of the sentences would then dictate the appropriate bracketings where they are once again distinguished with distinct powers of X. The total ILCs in the form of polynomials form the columns of Q, as before.
The first two basis verses 2:0 and 2:1 are already one-sentence-long. The
restriction of the third basis verse to its first sentence reduces its LC
to (11, 7, 2). The resulting word-delimited ALM sequences are:
2:0 (3, 4, 3) M ALL ALM ALM
2:1 (1, 1, 1) ALM
2:286 (11, 7, 2) LA L ALL A ALA A LA MA LA MA A
The first set of bracketings isolate each ALM-word:
2:0 M (ALL) (ALM) (ALM)
2:1 ALM
2:286 (LA) L (ALL) A (ALA) A (LA) (MA) (LA) (MA) A
The 2:1 sequence is already one word and requires no outermost bracketing.
The first word M in 2:0 can further be separated since the three
following words are related and thus grouped together:
2:0 M ( (ALL) (ALM) (ALM) )
The last two ALMs are similarly related and are grouped together:
2:0 M ( (ALL) ((ALM) (ALM)) )
The letters A and L in ALMs are adjacent in the original sentence and form
subwords that are separated from M by intervening non-ALM letters,
implying bracketing (AL)M denoted by AL.M:
2:0 M ( (ALL)
((AL.M) (AL.M)) )
The three letters in the second word ALL are
adjacent and a selection of a bracketing between AL.L and A.LL cannot be
made based on the textual structure. Both will have to be considered,
leading to the following two fully bracketed MuLCs:
W1 = M ( (AL.L) ((AL.M) (AL.M)) )
W2 = M ( (A.LL) ((AL.M) (AL.M)) )
with the following as the 2:0 total ILC:
U + W1 X + W2 X2,
where AdLC U = (3, 4, 3).
Basis verse 2:1 is only one word
with all adjacent letters, leading to two MuLCs: W1 =
AL.L and W2 = A.LL. The total ILC is:
U + W1 X + W2 X2,
where
AdLC U = (1, 1, 1). The third basis verse 2:286, restricted to its
first statement has AdLC U = (11, 7, 2) and contains 11 ALM-words of
lengths one, two and three. The 9th word in the sentence
contains no ALM letters. Thus, the first eight consecutive ALM-words are
separated from the last successive three, leading to the following
bracketing at the outset:
(LA L ALL A ALA A LA MA) (LA MA
A)
Isolating each word gives:
( (LA) L (ALL) A (ALA) A
(LA) (MA) ) ( (LA) (MA) A )
Further bracketings based similarly
on the words relations and linkages, propositions and phrases give the
following groupings:
( (((((LA) L) (ALL)) A) ((ALA) A)) ((LA) (MA)) ) ( (LA) ((MA) A) )
The two 3-words ALL and ALA each gets two bracketings: AL.L, A.LL
and AL.A, A.LA that are not determined by textual structures. The
resulting four fully bracketed MuLCs are denoted by Wi,
for 1 ≤ i ≤ 4, give the total ILC for 2:286:
U + W1 X + W2 X2 +
W3 X3 + W4 X4
The eight fully bracketed MuLCs (Wi's) from the three basis vectors
are then evaluated in the left/right S1 LC algebras, (S1: restricted to
sentence one):
(2:0)
(L)
W1 = (3217, 1022, 8980)
W2 = (11037, 7851, 12165)
ILC = (11037 X2 + 3217 X + 3,
7851 X2 + 1022 X + 4,
12165 X2 + 8980 X + 3)
(R)
W1 = (4330, 255, 7222)
W2 = (4472, 8491, 11688)
ILC = (4472 X2 + 4330 X + 3,
8491 X2 + 255 X + 4,
11688 X2 + 7222 X + 3)
(2:1)
(L)
W1 = (1349, 4347, 12435)
W2 = (3071, 1268, 8109)
ILC = (3071 X2 + 1349 X + 1,
1268 X2 + 4347 X + 1,
8109 X2 + 12435 X + 1)
(R)
W1 = (7575, 7065, 4633)
W2 = (5347, 615, 7616)
ILC = (5347 X2 + 7575 X + 1,
615 X2 + 7065 X + 1,
7616 X2 + 4633 X + 1) right
(2:286)
(L)
W1 = (3686, 6454, 7956)
W2 = (10872, 8387, 11693)
W3 = (4554, 8478, 5805)
W4 = (319, 3164, 8254)
ILC = (319 X4 + 4554 X3 +
10872 X2 + 3686 X + 11,
3164 X4 + 8478 X3 +
8387 X2 + 6454 X + 7,
8254 X4 + 5805 X3 +
11693 X2 + 7956 X + 2)
(R)
W1 = (4263, 9504, 579)
W2 = (6137, 1585, 2832)
W3 = (9644, 8252, 6333)
W4 = (1400, 6575, 7043)
ILC = (1400 X4 + 9644 X3 +
6137 X2 + 4263 X + 11,
6575 X4 + 8252 X3 + 1585 X +
9504 X + 7,
7043 X4 + 6333 X3 +
2832 X2 + 579 X + 2)
Once, the ILCs of the basis verses are known they form the columns of the Q matrix as previously defined. The action of Q on the versors leads to external LCs (XLCs) associated with the versors. The XLCs are vectors with polynomial components. The products of the LCs are defined through Q. The vectors are first transferred to the left/right (co)bif algebras via the inverse of Q, Q−1, multiplied there, then transferred back to the LC vector space via Q again turning the vector space into an algebra: UV = Q(Q−1(U) ⋅ Q−1(V)), where U and V are two arbitrary LCs. Initially, the entries in Q were all numbers (from F12597) and so was the inverse, Q−1. However, when the entries in Q are polynomials, the corresponding entries in Q−1 are, in general, rational functions of X, that is ratios of polynomials: f(X)/g(X), with f(X) and g(X) in F12697[X]. Moreover, the polynomials degrees can also grow unbounded (through multiplications). To confine the matrix operations to polynomials and at the same time impose a bound on the degrees of all polynomials, methods from modular arithmetic will be appropriated.
In modular arithmetic numbers wrap around a fixed value (the modulus), effectively treating arithmetic as movement around a circle. In modulus n, there are n numbers, {0, 1, 2, ... , n − 1}, arranged on the circle where the sequence repeats. In this system, n coincides with zero: x + n ≡ n (mod n) and x ⋅ n ≡ n, for all x, (n behaves exactly like zero: x + 0 = x, x ⋅ 0 = 0). The addition and multiplication simply continue and wrap around the circle. Given an arbitrary integer m, its location on the circle can be readily obtained. Let m = q ⋅ n + r, with quotient q and remainder 0 ≤ r < n. Then m wraps around the circle q times (q can be zero for m < n) until it settles at r. The number of times it wraps around (the quotient q) is insignificant. The location of m on the circle is solely determined by the remainder r. Thus all integers with the same remainder r are assigned to the same location on the circle and hence are considered equivalent. This arithmetic system is denoted as ℤ/n or ℤn (with n elements; 0 through n − 1), where ℤ is the set of all integers. A familiar example is the clock arithmetic (mod 12). It should be noted that if n is a composite number then it has a nontrivial factorization: n = ab for a, b ∈ ℤ/n with a, b > 1. Since, n ≡ 0 (mod n) this implies ab ≡ 0 (mod n) making a and b zero-divisors; two nonzero numbers with zero product. This can only be avoided if n is a prime number. For a prime p, the finite ring ℤ/p is a field and is denoted as Fp. In fields, all nonzero elements have multiplicative inverses.
The system of modular arithmetic is not limited to the set of integers. It equally applies to polynomials as well. Let ℤ[X] denote the polynomials in X with integer coefficients. Then ℤ/n[X] is the set of polynomials where modular arithmetic is only applied to the coefficients. In other words, the coefficients come from ℤ/n. However, there are no restrictions on the powers of X (can be any positive integer). Therefore, there are infinitely many such polynomials. In a more general setting, the modular arithmetic applies to both coefficients and the powers via polynomial division (modulo a fixed polynomial). Like integers, polynomials have quotients and remainders when divided by another. A fixed polynomial p(X) can serve as the modulus, similar to n with integers. Just as the integers with the same remainder by n are equivalent (≡ (mod n)), all polynomials with the same remainder when divided by p(X) are considered equivalent (≡ (mod p(X))). If f(X) is a polynomial, it can be expressed as f(X) = q(X) p(X) + r(X), with quotient q(X) and remainder r(X). Note that the quotient can be zero if the degree of f(X) is less than that of p(X) and f(X) would then be the same as the remainder. The remainder r(X) is always of smaller degree than the divisor p(X). The sum f(X) + p(X) = (q(X) + 1) p(X) + r(X) has the same remainder r(X) and therefore is equivalent to f(X). This implies that p(X) is behaving exactly like the zero-polynomial analogous to the behavior of n with integers. Let s(X) and t(X) be two equivalent polynomials: s(X) ≡ t(X) (mod p(X)). Then, s(X) = q1(X) p(X) + r(X) and t(X) = q2(X) p(X) + r(X), with quotients q1(X) and q2(X) and the same remainder r(X). Then, their difference, s(X) − t(X) = (q1(X) − q2(X)) p(X), is a multiple of p(X). This implies that equivalent polynomials differ by multiples of p(X) and a further indication that the modulus polynomial p(X) behaves like zero. The algebraic structure of these polynomials perfectly mirrors the integers modulo n, with the polynomial p(X) taking over the exact role of the integer n. Since, the remainder polynomials are of lower degrees than the divisor polynomial p(X), the degrees of polynomials are always less than that of p(X) and hence bounded. The resulting set of such polynomials modulo p(X) with integers modulo n coefficients is denoted by ℤn[X]/p(X). The bounds on both the coefficients and the degrees ensures a finite number of such polynomials.
In ℤ/n, the primality of n determines whether there are zero-divisors. It also determines whether all non-zero elements have multiplicative inverses (reciprocals). The modulus polynomial p(X) in ℤn[X]/p(X) plays the same role. The analogous notion is that of irreducible polynomials. They are the polynomial equivalents of prime numbers. Just as a prime number cannot be broken down into smaller factors, an irreducible polynomial cannot be factored into simpler, non-constant polynomials. In Fp[X]/f(X) with irreducible f(X) there are no zero-divisors and all non-zero polynomials have multiplicative inverses or reciprocals. This is clearly not true for ordinary polynomials since the reciprocal of a non-constant polynomial is not a polynomial. This is similar to the integers which do not have reciprocals (with the trivial exceptions of 1 and −1).
All polynomials in F12697[X]/p(X) are of lower degrees than
that of the divisor polynomial p(X). The polynomial expressions in the
ILCs of 2:282 and the basis verses are of degrees up to four. This
suggests setting the bound on the degrees of all polynomials to a maximum
of four. Accordingly, the degree of p(X) will be set to five, limiting
the remainders to degrees no more than four:
p(X) = c0 + c1 X +
c2 X2 + c3 X3 +
c4 X4 + c5 X5.
The modulus polynomial functions as the zero-polynomial in
F12697[X]/p(X): p(X) ≡ 0 (mod p(X)). This
establishes X5 in terms of the lower powers of X:
X5 = − (c0 + c1 X +
c2 X2 + c3 X3 +
c4 X4) / c5 = −
(c0/c5 + c1/c5 X +
c2/c5 X2 +
c3/c5 X3 +
c4/c5 X4).
Higher powers of X can similarly be expressed as linear combinations
of elements in {1, X, X2, X3, X4}. For
instance, the next power X6 is obtained as follows:
X6 = X X5
=
X (− c0/c5 −
c1/c5 X −
c2/c5 X2 −
c3/c5 X3 −
c4/c5 X4)
= − (c0/c5 X +
c1/c5 X2 +
c2/c5 X3 +
c3/c5 X4 +
c4/c5 X5)
= − (c0/c5 X +
c1/c5 X2 +
c2/c5 X3 +
c3/c5 X4) +
c4/c5 (c0/c5 +
c1/c5 X +
c2/c5 X2 +
c3/c5 X3 +
c4/c5 X4)
=
c0c4/c52 +
(c1c4/c52 −
c0/c5) X +
(c2c4/c52 −
c1/c5) X2 +
(c3c4/c52 −
c2/c5) X3 +
(c42/c52 −
c3/c5) X4
On the right hand side the occurrences of X5 have been
replaced with its equivalent expression of lower powers of X. All
higher powers Xn (n ≥ 5) can similarly be expressed
in polynomials of degrees four or less.
The choice of p(X) directly dictates the algebraic structure of the
resulting polynomials since it specifies how the higher powers of X (deg
> 4) are expressed as linear combinations of the lower powers of X
(deg ≤ 4). To keep the computations efficient and manageable the
simplest choice will be made were X5 is defined in terms of a
linear polynomial a + b X. A particular choice would be
X5 = X − 2 since p(X) = X5 − X + 2
is irreducible in F12697[X].
This results in a system of polynomials,
F12697[X]/(X5 − X + 2), where all non-zero
polynomials have reciprocals, forming a field. As an example, the
multiplicative inverse of X, X−1 = 1/X can be computed
as follows:
X5 = X − 2
Dividing both
sides by X gives:
X4 = 1 − 2 X−1
Which then gives
X−1 = 1/2 (1 − X4)
The transport-of-structure matrix Q(X) consists of entries in
F12697[X]/(X5 − X + 2). The rational function
entries in Q−1(X) have now turned into their polynomial
equivalents since the ratios of polynomials are again, polynomials. Let
Q−1(X) = (qij(X)), where qij(X) is
the polynomial entry at row i and column j. Their left/right
forms are:
(L)
q11(X) = 3905 X4 + 1104 X3 +
5871 X2 + 5240 X + 1793
q12(X) = 579 X4 + 9576 X3 +
2165 X2 + 10251 X + 7276
q13(X) = 7930 X4 + 12030 X3 +
6226 X2 + 11760 X + 12301
q21(X) = 11516 X4 + 9310 X3 +
10389 X2 + 8443 X + 7196
q22(X) = 11765 X4 + 2226 X3 +
1664 X2 + 12309 X + 3628
q23(X) = 7787 X4 + 8516 X3 +
12337 X2 + 232 X + 5329
q31(X) = 11130 X4 + 7442 X3 +
11631 X2 + 12179 X + 4496
q32(X) = 9046 X4 + 3799 X3 +
11535 X2 + 5305 X + 5017
q33(X) = 4892 X4 + 7702 X3 +
5301 X2 + 4136 X + 366
(R)
q11(X) = 4767 X4 + 8799 X3 +
761 X2 + 5377 X + 6479
q12(X) = 4155 X4 + 4860 X3 +
11030 X2 + 11589 X + 10249
q13(X) = 9494 X4 + 12616 X3 +
6583 X2 + 10896 X + 8359
q21(X) = 492 X4 + 1447 X3 +
6480 X2 + 6539 X + 5174
q22(X) = 2563 X4 + 5622 X3 +
6234 X2 + 4104 X + 8574
q23(X) = 1038 X4 + 5825 X3 +
6720 X2 + 5316 X + 970
q31(X) = 3141 X4 + 12556 X3 +
1472 X2 + 6714 X + 5706
q32(X) = 887 X4 + 9736 X3 +
12622 X2 + 5810 X + 11709
q33(X) = 7841 X4 + 3000 X3 +
7802 X2 + 10131 X + 1514
With matrices Q(X) and its inverse Q−1(X) given, the
left/right (co)bif algebra structures can be transported to the
corresponding left/right LC vector spaces over
F12697[X]/(X5 − X + 2). The general form of
the resulting multiplication table of the S1 LC algebras is:
AA = aaa A + aal L + aam M
AL = ala A + all L + alm M
AM = ama A + aml L + amm M
LA = laa A + lal L + lam M
LL = lla A + lll L + llm M
LM = lma A + lml L + lmm M
MA = maa A + mal L + mam M
ML = mla A + mll L + mlm M
MM = mma A + mml L + mmm M
The 27 structure constants aaa
through mmm are polynomials in X.
Their left/right forms are:
(L)
aaa = 1306 X4 +
10958 X3 + 12081 X2 + 7818 X + 2726
aal = 7144 X4 +
11676 X3 + 1073 X2 + 8349 X + 90
aam = 7847 X4 +
6994 X3 + 10698 X2 + 11712 X + 5492
ala = 3108 X4 +
7181 X3 + 10617 X2 + 1304 X + 5188
all = 4966 X4 +
6895 X3 + 12241 X2 + 7275 X + 10276
alm = 552 X4 +
11770 X3 + 9358 X2 + 3809 X + 229
ama = 9039 X4 +
308 X3 + 3269 X2 + 7270 X + 3095
aml = 7412 X4 +
2140 X3 + 12139 X2 + 7754 X + 4857
amm = 822 X4 +
9031 X3 + 5752 X2 + 70 X + 5655
laa = 4374 X4 +
3690 X3 + 7672 X2 + 2540 X + 12554
lal = 10561 X4 +
7145 X3 + 6744 X2 + 1036 X + 5050
lam = 11681 X4 +
6643 X3 + 912 X2 + 2559 X + 3010
lla = 10382 X4 +
8451 X3 + 12068 X2 + 8910 X + 6712
lll = 6322 X4 +
4751 X3 + 7510 X2 + 11093 X + 1183
llm = 2855 X4 +
8526 X3 + 6569 X2 + 2166 X + 6306
lma = 6052 X4 +
1362 X3 + 8535 X2 + 1371 X + 2324
lml = 11458 X4 +
332 X3 + 5045 X2 + 250 X + 11602
lmm = 8842 X4 +
5749 X3 + 2971 X2 + 1719 X + 7640
maa = 10252 X4 +
170 X3 + 375 X2 + 4344 X + 1200
mal = 91 X4 +
7606 X3 + 11488 X2 + 11891 X + 8589
mam = 12623 X4 +
9350 X3 + 3917 X2 + 500 X + 3834
mla = 12114 X4 +
8749 X3 + 2884 X2 + 1517 X + 11023
mll = 6002 X4 +
6219 X3 + 6369 X2 + 5163 X + 356
mlm = 6382 X4 +
4916 X3 + 12044 X2 + 11377 X + 8086
mma = 6156 X4 +
1322 X3 + 10376 X2 + 7633 X + 1306
mml = 4318 X4 +
12209 X3 + 9263 X2 + 8004 X + 4230
mmm = 8931 X4 +
12409 X3 + 10039 X2 + 6968 X + 6763
(R)
aaa = 1572 X4 +
11581 X3 + 3069 X2 + 4977 X + 4164
aal = 7796 X4 +
10432 X3 + 8299 X2 + 9131 X + 7995
aam = 7307 X4 +
7348 X3 + 12350 X2 + 7755 X + 6656
ala = 5085 X4 +
10721 X3 + 7957 X2 + 4055 X + 4573
all = 4163 X4 +
1449 X3 + 3754 X2 + 11129 X + 11329
alm = 6596 X4 +
1627 X3 + 7757 X2 + 2608 X + 4822
ama = 11581 X4 +
3715 X3 + 2213 X2 + 3441 X + 1475
aml = 5007 X4 +
4267 X3 + 5507 X2 + 11634 X + 2516
amm = 1815 X4 +
5197 X3 + 206 X2 + 4342 X + 7728
laa = 11624 X4 +
11111 X3 + 6913 X2 + 946 X + 1404
lal = 10122 X4 +
5847 X3 + 12246 X2 + 9497 X + 11658
lam = 9486 X4 +
5674 X3 + 9764 X2 + 5973 X + 11704
lla = 2826 X4 +
2869 X3 + 741 X2 + 9860 X + 232
lll = 11995 X4 +
5533 X3 + 8902 X2 + 4157 X + 2482
llm = 5865 X4 +
946 X3 + 3722 X2 + 8734 X + 3874
lma = 3523 X4 +
9326 X3 + 2112 X2 + 10676 X + 2878
lml = 10629 X4 +
1740 X3 + 2257 X2 + 5511 X + 2401
lmm = 5057 X4 +
10703 X3 + 2629 X2 + 8482 X + 11083
maa = 9194 X4 +
2878 X3 + 12012 X2 + 6415 X + 10168
mal = 6359 X4 +
11840 X3 + 8822 X2 + 1664 X + 5792
mam = 8173 X4 +
9596 X3 + 3822 X2 + 5191 X + 1642
mla = 705 X4 +
10841 X3 + 6830 X2 + 2041 X + 9390
mll = 8315 X4 +
7691 X3 + 404 X2 + 12302 X + 98
mlm = 9427 X4 +
2343 X3 + 7957 X2 + 1134 X + 2727
mma = 2763 X4 +
1013 X3 + 839 X2 + 7902 X + 4166
mml = 6664 X4 +
6202 X3 + 2382 X2 + 10899 X + 11663
mmm = 7716 X4 +
4640 X3 + 4756 X2 + 1457 X + 7049
The original LC algebras had underlying vector spaces of dimension three with scalars from the finite field F12697. There were a total of 126973 vectors. In the new LC algebras the field F12697 (regarded as a one-dimensional vector space over itself with a singleton basis {1 = X0}) is replaced with its more elaborate left/right, extension fields F12697[X]/(X5 − X + 2) which are five-dimensional vector spaces with basis {1, X, X2, X3, X4}. Subsequently, there are 126975 polynomials of degrees up to four. As they form the three components of the basis letters {A, L, M}, there are (126975)3 = 1269715 vectors in the polynomial-based LC algebras. The X-dependencies of the pairwise products of the basis letters in turn imply the indirect dependencies of the basis letters on X.