Linear content (LC) algebras were constructed for verse 2:282. The
unital algebras were the principal isotope (𝕊, ∘) and
its mirror dual (𝕋, *) with respect to the invertible LC
U = 107 A + 66 L + 31 M = (107, 66, 31). The isotopes were
subsequently merged into an encompassing algebra 𝕄 consisting
of ordered-pairs of vectors Z = (X, Y). The addition was
componentwise and the multiplication unified the two isotope
products (∘ and *) with vectors crossed:
Z1Z2 =
(X1, Y1)(X2, Y2) =
(X1 ∘ Y2,
Y1 * X2)
The combined algebra had the right multiplicative identity that also
was the left exchange involution. The two isotopes, nevertheless can
be consolidated in many different ways. Drawing on the dual symmetry
between the two isotopes allows a merger that is more intrinsic and
proper. Recall, the isotopes are symmetrically linked via swapping
the left and right inverse multiplication maps
LU−1(•),
RU−1(•).
Let ℳ be the mirror operator that swaps the inverse maps (thus,
swapping 𝕊 and 𝕋) while the
identity operator ℐ leaves them intact:
ℐ𝕊 = 𝕊
ℐ𝕋 = 𝕋
ℳ 𝕊 = 𝕋
ℳ 𝕋 = 𝕊
Let ℳ2 = ℳℳ be the composition of ℳ with
itself; that is it is applied twice in a row. Then,
ℳ2 𝕊 = ℳ(ℳ 𝕊)
= ℳ 𝕋
= 𝕊
And similarly
ℳ2 𝕋 = ℳ(ℳ 𝕋)
= ℳ 𝕊
= 𝕋
The actions of ℳ2 are identical to those of ℐ.
Therefore, ℳ2 = ℐ. Since, ℐ acts trivially,
its composition with itself is also trivial: ℐ2 =
ℐ. Its pre or post compositions with ℳ do not alter the
actions of ℳ: ℐℳ = ℳℐ = ℳ. It follows
that the set G = {ℐ, ℳ} is closed under
composition. It forms an abelian group of order two. The order of a group
is the number of its elements. This is the smallest nontrivial group. A
group of order one is just the trivial group {ℐ} consisting of a
single identity element and hence, with trivial action.
Similar to magma, the group G can also be associated with
its group algebra over some field, for instance, over the Real numbers:
ℝ[G]. Its elements are linear operators that are
formal linear combinations of ℐ and ℳ:
g = x ℐ + y ℳ for x, y ∈ ℝ. Two such
elements g1 = x1 ℐ + y1 ℳ
and g2 = x2 ℐ + y2 ℳ
can be added the obvious way:
g1 + g2 =
(x1 ℐ + y1 ℳ) +
(x2 ℐ + y2 ℳ)
=
(x1 + x2) ℐ +
(y1 + y2) ℳ
The formal sums x ℐ + y ℳ
form a two-dimensional vector space with G
as the basis. The standard basis vectors are (1, 0) and (0, 1) standing
for ℐ and ℳ, resp. In standard basis g1 and
g2 are:
g1 = x1 (1, 0) + y1 (0, 1)
=
(x1, 0) + (0, y1)
=
(x1, y1)
Similarly,
g2 = (x2, y2).
The addition in standard basis is straightforward:
g1 + g2 =
(x1, y1) + (x2, y2)
=
(x1 + x2, y1 + y2)
The multiplication is defined as follows:
g1g2 =
(x1, y1)(x2, y2)
=
(x1 ℐ + y1 ℳ)(x2 ℐ + y2 ℳ)
=
x1x2 ℐ2 +
x1y2 ℐℳ +
y1x2 ℳℐ +
y1y2 ℳ2
=
x1x2 ℐ + x1y2 ℳ +
y1x2 ℳ + y1y2 ℐ
=
(x1x2 + y1y2) ℐ +
(x1y2 + y1x2) ℳ
=
(x1x2 + y1y2,
x1y2 + y1x2)
With the multiplication so defined the group algebra ℝ[G] is equivalent to the algebra of split complex numbers also known as the hyperbolic numbers. These are numbers of the form x + jy with x, y ∈ ℝ, and the hyperbolic unit j ∉ ℝ and j2 = 1, that is j ≠ ±1. This way, two copies of ℝ (one-dimensional algebras) are combined to form a two-dimensional number system with the prescribed multiplication. The operators ℐ and ℳ correspond directly to 1 and j, resp. Likewise, an appropriately constructed group algebra of G allows the merger of the two isotopes 𝕊 and 𝕋 into a larger six-dimensional algebra 𝕄.
Since both isotopes have the same underlying vector space, an algebra
𝔹 can be constructed over the same vector space but with both
isotope-multiplications simultaneously: (𝔹, ∘, *). Both
isotopes 𝕊 and 𝕋 are present in 𝔹 as reducts by forgetting
an operation (multiplication) from 𝔹. Let 𝔹[G]
be its group algebra. Its elements are operators of the form
g = Xℐ + Yℳ with vectors X, Y ∈ 𝔹.
The sums and products are defined identically to those of the
split complex numbers:
g1 + g2 =
(X1, Y1) + (X2, Y2)
=
(X1 + X2, Y1 + Y2)
The product g1g2 is determined via
its action on 𝕊 and 𝕋.
Let A = {𝕊, 𝕋} and 𝔸 ∈
A. Then,
ℐ𝔸
= 𝔸
ℳ 𝔸= 𝔸'
ℳ 𝔸' = 𝔸
where 𝔸 and 𝔸', distinct
elements in A, stand for the two isotopes, one for
𝕊 and the other for 𝕋. Throughout, the juxtaposition
XiYj
(for Xi, Yj ∈ 𝔹)
represents a generic product
Xi □𝓀 Yj,
where the choice of □𝓀 ∈ {∘, *}
for 𝓀 = 1,2 is independent for each occurrence of
juxtaposition. The action of
g1g2 on 𝔸 is as follows:
(g1g2) 𝔸 = g1(g2 𝔸)
=
(X1, Y1)((X2, Y2) 𝔸)
=
(X1 ℐ + Y1 ℳ)((X2 ℐ + Y2 ℳ) 𝔸)
=
(X1 ℐ + Y1 ℳ)(X2 ℐ 𝔸 + Y2 ℳ 𝔸)
=
(X1 ℐ + Y1 ℳ)(X2 𝔸 + Y2 𝔸')
=
X1X2 ℐ 𝔸 +
X1Y2 ℐ 𝔸' +
Y1X2 ℳ 𝔸 +
Y1Y2 ℳ 𝔸'
=
X1X2 𝔸 +
X1Y2 𝔸' +
Y1X2 𝔸' +
Y1Y2 𝔸
=
(X1X2 + Y1Y2) 𝔸 +
(X1Y2 + Y1X2) 𝔸'
Setting 𝔸 = 𝕊 gives 𝔸' = 𝕋:
(g1g2) 𝕊 =
(X1X2 + Y1Y2) 𝕊 +
(X1Y2 + Y1X2) 𝕋
The expression on the RHS indicates that the operations
in the first bracket
(X1X2 + Y1Y2) are to
be carried out in (𝕊, ∘) and the second bracket
(X1Y2 + Y1X2)
in (𝕋, *). The generic product will now be explicitly set
to the products in the respective isotopes:
(g1g2) 𝕊 =
(X1 ∘ X2 +
Y1 ∘ Y2) 𝕊 +
(X1 * Y2 + Y1 * X2) 𝕋
=
(X1 ∘ X2 +
Y1 ∘ Y2) ℐ 𝕊 +
(X1 * Y2 + Y1 * X2)
ℳ 𝕊
Factoring 𝕊 out from both sides gives:
g1g2 =
(X1 ∘ X2 +
Y1 ∘ Y2) ℐ +
(X1 * Y2 + Y1 * X2) ℳ
=
(X1 ∘ X2 +
Y1 ∘ Y2)(1, 0) +
(X1 * Y2 + Y1 * X2)(0, 1)
=
(X1 ∘ X2 +
Y1 ∘ Y2, 0) +
(0, X1 * Y2 + Y1 * X2)
=
(X1 ∘ X2 +
Y1 ∘ Y2,
X1 * Y2 + Y1 * X2)
where, ℐ and ℳ are replaced with their standard
basis representations (1, 0) and (0, 1). The two isotopes (𝕊,
∘) and (𝕋, *) are thus merged to form a larger algebra
𝕄 whose elements are ordered pairs Z = (X, Y) with a unified
multiplication mirroring that of the split complex numbers.
The isotopes are unital algebras with a common
multiplicative identity e = U2.
Let E = (e, 0)
and J = (0, e).
For Z = (X, Y) ∈ 𝕄
EZ = (e, 0)(X, Y)
=
(e ∘ X + 0 ∘ Y,
e * Y + 0 * X)
= (X, Y)
= Z
And
ZE = (X, Y)(e, 0)
=
(X ∘ e + Y ∘ 0,
X * 0 + Y * e)
= (X, Y)
= Z
E is,
therefore the multiplicative identity. Accordingly, it will be denoted
by 𝟙.
J2 = (0, e)
(0, e)
=
(0 ∘ 0 +
e ∘ e,
0 * e +
e * 0)
=
(e, 0)
=
𝟙
It is idempotent and, hence the hyperbolic unit. It will be denoted
by 𝕁.
𝕁Z =
(0, e)(X, Y)
=
(0 ∘ X + e ∘ Y,
0 * Y + e * X)
= (Y, X)
Z𝕁 =
(X, Y)(0, e)
=
(X ∘ 0 + Y ∘ e,
X * e + Y * 0)
= (Y, X)
𝕁 is the
exchange involution and commutes with all elements:
𝕁Z =
Z𝕁
for all Z ∈ 𝕄. Furthermore,
(X, 0)(X, 0) =
(X ∘ X, 0)
(0, Y)(0, Y) =
(Y ∘ Y, 0)
(X, 0)(0, Y) =
(0, X * Y)
(0, Y)(X, 0) =
(0, Y * X)
Elements with one zero coordinate behave as vectors in the isotopes.
That is, a vector V ∈ 𝔸 can be represented as an element in
𝕄 as (V, 0) or (0, V). The
products of such elements are purely in 𝕊 or in 𝕋.
Z = (X, Y)
=
(X, 0) + (0, Y)
=
(X, 0) +
𝕁(Y, 0)
This concludes the parallel to the split complex numbers
z = x + jy.
Let E = 1/2
(𝟙 +
𝕁) and
E* = 1/2
(𝟙 −
𝕁). Then
E2 = 1/4
(𝟙 +
𝕁)2
= 1/4
(𝟙2 +
2 𝟙𝕁 +
𝕁2)
= 1/4
(𝟙 + 2
𝕁 +
𝟙)
= 1/2
(𝟙 +
𝕁)
= E
And similarly
(E*)2 = 1/4
(𝟙 −
𝕁)2
= 1/4
(𝟙2 −
2 𝟙𝕁 +
𝕁2)
= 1/4
(𝟙 − 2
𝕁 +
𝟙)
= 1/2
(𝟙 −
𝕁)
=
E*
Both E and E*
are thus idempotent. Additionally,
EE* = 1/4
(𝟙 +
𝕁)
(𝟙 −
𝕁)
= 1/4
(𝟙2
−
𝕁2)
= 1/4
(𝟙 −
𝟙)
= 0
And similarly
E*E = 0
They are also zero-divisors.
𝕄 is a six-dimensional unital ALM LC algebra for verse 2:282 that is a merger of the two three-dimensional isotopes (𝕊, ∘) and (𝕋, *). The reflection symmetry between the two isotopes are first formalized with operators ℐ and ℳ whose appropriately constructed group algebra leads to a unified multiplication that naturally mirrors that of the split complex numbers. Moreover, the hyperbolic unit is the coordinate exchange involution that commutes with all elements in 𝕄. The idempotents and zero-divisors are derived from the multiplicative and the hyperbolic units. Algebra 𝕄 attains the left and right variants as before.