The linear-content (LC) algebras were constructed from the primary CoBif algebras via the bijective map Q : CoBif ⟶ LC and its inverse Q−1 : LC ⟶ CoBif. The algebra structures were transferred from basis {2:0, 2:1, 2:286} or {E1, E2, E3} to basis {A, L, M} also denoted as {F1, F2, F3}. Both CoBif and LC algebras are generic in the sense that they are not affiliated with any particular verse. It is, however, compelling to construct algebras explicitly associated with a given verse. To that end, algebras will be constructed specifically for verse 2:282.
Given a vector U in the LC algebra, it has its associated linear left and right multiplication maps: LU(X) = UX and RU(X) = XR. They map an arbitrary vector X by multiplication with U on the left and right resp. A vector is left (resp. right) invertible if its left (resp. right) map is bijective. That is distinct vectors are mapped to distinct vectors and any vector is a map of some unique vector. If a vector is both left and right invertible then it is simply called invertible. Given an invertible U, the multiplications in the LC algebras can be modified by leveraging the inverses of the left and right maps of U, while preserving the underlying vector space, resulting in an algebra that is directly associated with the vector U.
Algebras explicitly for verse 2:282 will be constructed based on its
internal LC (ILC): U = 107 A + 66 L + 31 M = (107, 66, 31). The left and
right linear maps LU(X) and RU(X) are completely
determined by their actions on the basis vectors, that is for X ∈
{A, L, M} in both left/right LC algebras:
(Left Generic LC Algebra)
LU(A) = UA =
(3903, 4238, 8589)
LU(L) = UL =
(7484, 349, 10226)
LU(M) = UM =
(10403, 3190, 2481)
RU(A) = AU =
(9263, 8030, 4438)
RU(L) = LU =
(2682, 11334, 8262)
RU(M) = MU =
(9908, 1938, 5426)
As bijective maps, they have inverses with actions that are
similarly determined via the basis:
LU−1(A) =
(9155, 11807, 8433)
LU−1(L) =
(6519, 562, 8580)
LU−1(M) =
(4822, 10348, 10747)
RU−1(A) =
(3832, 5331, 3982)
RU−1(L) =
(2017, 12229, 2484)
RU−1(M) =
(12107, 10482, 2243)
(Right Generic LC Algebra)
LU(A) = UA =
(8646, 7429, 3279)
LU(L) = UL =
(5386, 7058, 1134)
LU(M) = UM =
(12519, 5677, 9220)
RU(A) = AU =
(3975, 8593, 5412)
RU(L) = LU =
(7638, 6259, 919)
RU(M) = MU =
(9102, 3770, 7640)
And their inverses:
LU−1(A) =
(6763, 1440, 12347)
LU−1(L) =
(11767, 3783, 7379)
LU−1(M) =
(2532, 8031, 11047)
RU−1(A) =
(3456, 4386, 12377)
RU−1(L) =
(6244, 488, 7351)
RU−1(M) =
(1603, 7889, 9047)
Once, the actions of the maps and their inverses on the basis letters are
known the actions extend linearly to all vectors since every vector is a
unique linear combination of the basis vectors. The maps LU(X),
RU(X) and their inverses LU−1(X)
and RU−1(X) are consequently fully determined.
The new multiplication (∘) is defined in terms of the original
multiplication (⋅) in the generic LC algebras:
X ∘ Y = RU−1(X) ⋅
LU−1(Y)
The operands X and Y are first modified by the inverse maps of U,
(the ILC of 2:282), before being multiplied in the LC algebras. The
resulting left/right algebras are referred to as the principal isotopes,
(𝕊, ∘), with respect to the invertible element U. The
original multiplications are thus shifted and the algebras and the
structural behavior are localized at U. Isotopic algebras share the same
underlying structural properties while not being isomorphic. Isotopy is
thus more flexible than isomorphism.
The principal isotopes are also nonassociative as they are
defined in terms of the original nonassociative LC algebras.
However, unlike the original algebras, the principal isotopes are
unital. That is there exist a unique element (one for each isotope)
e = U2 that is the
multiplicative neutral element (the identity):
e ∘ v = v ∘ e = v,
for all v in the isotope. This is due to the cancellations
resulting from compositions of maps with their inverses
resulting in the trivial or the identity map. Recall, for
an arbitrary bijective map f(X)
f(X) ∘ f−1(X) =
f−1(X) ∘ f(X) = Id(X) = X.
The (∘) is the composition of functions:
f(X) ∘ g(X) = f(g(X)). The following shows how
e = U2 behaves
as the multiplicative identity in either isotope:
e ∘ v = RU−1(e)
⋅ LU−1(v)
= RU−1(U2)
⋅ LU−1(v)
= RU−1(RU(U))
⋅ LU−1(v)
= U ⋅ LU−1(v)
= LU(LU−1(v))
= v.
The first line is simply the definition of (∘) in terms of
(⋅) applied to the arguments e and v. In the
second line e is replaced by U2. In the
third line U2 is replaced by its equivalent form
U2 = U ⋅ U = RU(U). In the forth line U is
the result of cancellation of RU composed with its inverse
RU−1. The multiplication by U
on the left is then replaced in the fifth line by its
equivalent LU. The last line follows due to the
cancellation of LU composed with its inverse
LU−1. And similarly,
v ∘ e = RU−1(v)
⋅ LU−1(e)
= RU−1(v)
⋅ LU−1(U2)
= RU−1(v)
⋅ LU−1(LU(U))
= RU−1(v) ⋅ U
= RU(RU−1(v))
= v.
The third line uses the fact that U2 =
U ⋅ U = LU(U). In the next line LU
and its inverse LU−1 cancel out. The
multiplication on the right by U in the forth line is replaced by its
equivalent RU in the next line. The last line
follows from the cancellation of RU composed with its inverse
RU−1.
Thus e
behaves exactly like number 1 under
multiplication, 1 a = a 1 = a, for all numbers a.
The multiplication table (the structure constants)
for the left/right principal isotopes are obtained
from X ∘ Y = RU−1(X) ⋅
LU−1(Y) applied to the basis letters
{Fi, for 1 ≤ i ≤ 3}:
Fi ∘ Fj =
RU−1(Fi) ⋅
LU−1(Fj). Included also,
are the multiplicative identities for each isotope:
(2:282 Principal Isotope From Left LC Algebra)
AA = (2289, 10095, 3405)
AL = (9526, 11892, 11486)
AM = (9064, 7351, 889)
LA = (7158, 7900, 6408)
LL = (5257, 11476, 11256)
LM = (8396, 5047, 8539)
MA = (1455, 8722, 12143)
ML = (8521, 1869, 1354)
MM = (8819, 3340, 4891)
Multiplicative Identity: e = (2449, 4025, 7543)
(2:282 Principal Isotope From Right LC Algebra)
AA = (9626, 9472, 4735)
AL = (7414, 2662, 3822)
AM = (1951, 11472, 10906)
LA = (373, 1134, 8566)
LL = (8708, 12157, 9786)
LM = (7403, 1171, 4400)
MA = (5727, 9967, 10194)
ML = (10193, 2285, 8985)
MM = (662, 8628, 2558)
Multiplicative Identity: e = (5380, 1957, 485)
The infix multiplication sign (∘) between the basis
letters has been suppressed and the pairwise products are
implicitly displayed with juxtaposition as before.
The principal isotopes (𝕊, ∘) defined with X ∘ Y = RU−1(X) ⋅ LU−1(Y) have their mirror symmetric dual (𝕋, *) obtained via swapping the left and right inverses: X * Y = LU−1(X) ⋅ RU−1(Y) and with the same multiplicative identity e = U2. Other notable variations include reversal of the ordering of X and Y on the RHS. Altering the placement and order of maps along with arguments reversals and their combinations generate a family of distinct but closely related isotopes that localize structural properties at the invertible element U.
Isotopes (𝕊, ∘) and (𝕋, *) are closely linked.
Let G = LURU−1, then
G−1 = RULU−1.
Then, by definition
GX * G−1Y = LU−1
(GX) ⋅ RU−1
(G−1Y)
=
LU−1
(LURU−1(X)) ⋅
RU−1
(RULU−1(Y))
=
RU−1(X) ⋅
LU−1(Y)
= X ∘ Y
Note, the cancellations of LU and RU
with their inverses: LU−1LU
= Id = RU−1RU.
Replacing X with G−1X and Y with GY
gives the following equivalent form:
GG−1X * G−1GY =
G−1X ∘ GY
Which reduces to
X * Y = G−1X ∘ GY
It is beneficial
to have the two isotopes merged to form an encompassing algebra
𝕄 where the two components can interact systematically via a
single, unified product. The elements in 𝕄 are ordered pairs
of vectors (X, Y). Addition is defined componentwise:
(X1, Y1) + (X2, Y2) =
(X1 + X2 ,
Y1 + Y2)
Multiplication is defined with vectors crossed:
(X1, Y1)(X2, Y2) =
(X1 ∘ Y2,
Y1 * X2),
where the products
∘ and * on the RHS are split between the two coordinates.
The crossings between Xi's and Yj's provide
implicit linking between the two isotopes. Although, both 𝕊 and
𝕋 are unital with e as the common multiplicative
identity, there is no global (two-sided) identity in 𝕄.
Let E = (e, e). Then:
E(X, Y) = (e, e)(X, Y)
=
(e ∘ Y,
e * X)
= (Y, X)
Multiplication by E on the left simply swaps vectors in an
ordered pair. It is an involutive coordinate exchange.
If applied once more it returns the original pair:
E(E(X, Y)) = E(Y, X)
= (X, Y)
The multiplication by E on the right, however, is trivial:
(X, Y)E = (X, Y)(e, e)
=
(X ∘ e,
Y * e)
= (X, Y)
Thus, E is the left-exchange-involution and the right-identity.
In terms of the left/right maps of E they can be stated as
LE2 = RE = 𝟙. Additionally,
E2 = EE
=
(e, e)
(e, e)
=
(e ∘ e
, e * e)
=
(e, e)
= E
It is idempotent, an element whose square is the same as itself.
Consequently all higher powers of E are also equal to E regardless
of associativity. Another distinct feature of 𝕄 is the
presence of zero-divisors:
(X, 0)(Y, 0) =
(X ∘ 0,
0 * Y)
=
(0, 0)
(0, X)(0, Y) =
(0 ∘ Y,
X * 0)
=
(0, 0)
In particular, (X, 0)2 =
(0, 0) =
(0, Y)2.
These type of elements are called nilpotent; elements that become
zero when raised to some positive number. 𝕄 is therefore
partitioned into two zero-divisor spaces. Let 𝕌 be the
underlying vector space common to both 𝕊 and 𝕋.
Then, 𝕄 splits into two subspaces:
𝕄 = 𝕍 ⊕ 𝕎, with
𝕍 = 𝕌 ⊕ {0} =
{(X, 0) | X ∈ 𝕌} and
𝕎 = {0} ⊕ 𝕌 =
{(0, Y) | Y ∈ 𝕌}. The trivial
vector space {0} is of dimension zero and consists
of a single zero-vector 0 = (0, 0, 0) = 0 A + 0 L + 0 M
representing the null content. Within
subspaces, multiplication is trivial: for arbitrary
V1, V2 ∈ 𝕍 and
W1, W2 ∈ 𝕎,
V1V2 = (0, 0)
= W1W2.
The subspaces 𝕍 and 𝕎 can individually be regarded as mere
vector spaces since the products within each space is trivial.
Any vector space can be taken as a trivial algebra with zero product.
Both isotopes (𝕊, ∘) and (𝕋, *) are contained
(as isomorphic copies) in 𝕄. Let V ∈ 𝕍 and W ∈ 𝕎.
Then, VW = (X, 0)(0, Y) =
(X ∘ Y, 0)
which corresponds directly to (𝕊, ∘) and likewise
WV = (0, Y)(X, 0) =
(0, Y * X) that similarly corresponds to (𝕋, *).
An element X ∈ 𝕊 coincides with (X, 0) ∈
𝕍 and similarly Y ∈ 𝕋 lifts to
(0, Y) ∈ 𝕎.
Maps, like elements can be nilpotent as well. That is after
self-composition a number of times they become identically zero.
In particular, square-zero maps self-compose to zero. Let E =
E1 + E2 where
E1 = (e, 0) and
E2 = (0, e).
Their corresponding right multiplication maps are:
P1(B) = RE2(B) = BE2 and
P2(B) = RE1(B) = BE1,
for B = (X, Y) ∈ 𝕄. Then,
P1(B) = BE2
= (X, Y)E2
=
(X, Y)(0, e)
= (X, 0)
And similarly,
P2(B) = BE1
= (X, Y)E1
=
(X, Y)(e, 0)
= (0, Y)
Thus, Pi, for i ∈ {1, 2} returns the
ith coordinate of the ordered pair B = (X, Y):
P1(X, Y) = (X, 0) and
P2(X, Y) = (0, Y).
These are the projection operators. Both are idempotent:
P12 = P1,
P22 = P2 as can readily be verified.
Note, P1 + P2 = 𝟙, the trivial identity
operator. For B = (X, Y) ∈ 𝕄:
𝟙B = (P1 + P2)B
=
P1(X, Y) + P2(X, Y)
=
(X, 0)) + (0, Y)
= (X, Y)
= B
With E = (e, e) acting as
the left involutive coordinate exchange, let LE(B)
be the left multiplication map of E in 𝕄:
LE(B) = EB = E(X, Y) = (Y, X) for B = (X, Y) ∈ 𝕄.
The linear map d = P2LEP1
is square-zero:
dB = P2LEP1B
=
P2LEP1(X, Y)
=
P2LE(X, 0)
=
P2(0, X)
=
(0, X)
d2B = d(dB)
=
d(0, X)
=
P2LEP1(0, X)
=
P2LE(0, 0)
=
P2(0, 0)
=
(0, 0)
Since B is arbitrary, this implies d2 is the zero-map:
d2 = 0. Swapping P1 and P2
also gives similar square-zero map
h = P1LEP2:
hB = P1LEP2B
=
P1LEP2(X, Y)
=
P1LE(0, Y)
=
P1(Y, 0)
=
(Y, 0)
h2B = h(hB)
=
h(Y, 0)
=
P1LEP2(Y, 0)
=
P1LE(0, 0)
=
P1(0, 0)
=
(0, 0)
for arbitrary B = (X, Y) ∈ 𝕄, and hence h2 = 0.
The two operators d and h complement each other. Given B = (X, Y),
dB = (0, X) and hB = (Y, 0)
as shown above. Thus,
(d + h)B = dB + hB
=
(0, X) + (Y, 0)
= (Y, X)
= E(X, Y)
= EB
= LE(B)
Since B is arbitrary, this implies d + h = LE.
Furthermore,
dhB = d(h(B))
= d(Y, 0)
= (0, Y)
And similarly,
hdB = h(d(B))
= h(0, X)
= (X, 0)
Therefore,
(dh + hd)B = dhB + hdB
=
(0, Y) + (X, 0)
= (X, Y)
= B
Since, this is true for all B ∈ 𝕄, the sum must be
the trivial identity operator: dh + hd = 𝟙.
𝕄 is a six dimensional ALM LC algebra over the finite field F12697 that is associated with the verse 2:282. It is a merger of the principal isotope 𝕊 and its mirror dual 𝕋, the two localizations of the LC algebras at U, the invertible internal linear ALM content of 2:282. Its elements (the long or double vectors) are ordered pairs of vectors (X, Y), with X and Y as linear contents (a, l, m) with a, l, m ∈ F12697. Although, both isotopes are unital algebras with common multiplicative identity e = U2, 𝕄 retains only the right-identity E = (e, e) that also is the left involutive coordinate exchange. Other notable features are the presence of idempotent, zero-divisors and nilpotent elements and maps. Particularly noteworthy is the construction of a pair of complementary nilpotent maps d and h, (d2 = h2 = 0), whose sum is an involution: (d + h)2 = dh + hd = 𝟙. Since, the isotopes are derived from both left and right generic LC algebras, 𝕄 also acquires left and right variants.