Analysis of Complex Entities in Algebra B

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Analysis of Complex Entities in Algebra B

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Alejandro Jesús Bermejo Valdés^*

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Submitted:

19 June 2024

Posted:

20 June 2024

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Abstract

We present an exploration of algebra B, a recently published unital and non-associative algebra. Unique complex entities emerged from this algebra, distinct from both quaternion and complex number systems, which we termed treons. We defined the treonic number system and established an isomorphism between this system and the real vector space. Our findings revealed that the treonic representation maintained structural integrity under the defined operations. Based on this foundation, we proceed to determine Euler identity for algebra B within the treonic system. By presenting the fundamental definitions and properties of algebra B, we derived a generalized version of Euler identity applicable within this algebra. This formula revealed the emergence of hyperbolic trigonometric entities, extending the applicability of Euler identity beyond traditional complex numbers. Our results provide a theoretical foundation for a deeper understanding of the properties and behaviors of complex entities within this expanded algebraic framework, thus enabling new theoretical developments and practical applications in the realms of advanced mathematics and theoretical physics.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

Introduction

The algebra B is a recently described algebraic structure by Alejandro Bermejo [1]; it is defined as a non-associative and unital algebra with the potential to satisfy the definitional requirements of Lie and Malcev algebras when the respective products of these latter algebras are defined using the product of algebra B. However, an added quality of this algebra is the emergence of complex entities with a structure different from existing ones. They appear similar to quaternions [2] lacking one dimension, but they are not. They also resemble the set of complex numbers

C^{2}

[3] with an additional dimension, but they are not that either. Our study aims to investigate the structural properties of these entities, which we refer to as "treons", and to establish an isomorphism between algebra B and the treonic number system. We will demonstrate the mathematical correctness of representing elements of algebra B in the form

a_{1} + a_{2} i + a_{3} j

, ultimately proving the existence of an isomorphism to

R^{3}

. Based on this, that is, on the possibility of expressing the elements of algebra B in the form of treons, we seek to explore how Euler’s identity manifests in the context of algebra B. Euler’s identity (or Euler’s formula),

e^{i θ} = cos θ + i sin θ

, is a cornerstone of complex analysis, illustrating the profound relationship between exponential and trigonometric functions [4,5]. We find that the structure of this identity in algebra B has a similar form, but with its own unique characteristics and differences.

1. Products in Quaternion and $C^{2}$ Algebras

The product of quaternions is defined as [2]:

(a_{1}, a_{2}, a_{3}, a_{4}) (b_{1}, b_{2}, b_{3}, b_{4}) \equiv

(a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3} - a_{4} b_{4}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{4} - a_{4} b_{3}, a_{1} b_{3} - a_{2} b_{4} + a_{3} b_{1} + a_{4} b_{2}, a_{1} b_{4} + a_{2} b_{3} - a_{3} b_{2} + a_{4} b_{1}) .

And the product in

C^{2}

is defined as [3,6]:

(a_{1}, a_{2}) (b_{1}, b_{2}) \equiv (a_{1} b_{1} - a_{2} b_{2}, a_{1} b_{2} + a_{2} b_{1}) .

2. Product in Algebra B

The product ⊙ in algebra B is defined as [1]:

(a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) \equiv (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}) .

Both the quaternionic product and the product in

C^{2}

are different from the product in B. It is not sufficient to nullify one or another component to derive algebra B from quaternions. Similarly, it does not make sense to impose an additional dimension to

C^{2}

to reach the product of algebra B. Therefore, from algebra B, the complex elements that arise must be different from the complex elements of the set

C^{2}

and the quaternions.

3. Representations in Quaternion and $C^{2}$ Algebras

Quaternions can be represented as:

(a_{1}, a_{2}, a_{3}, a_{4}) = a_{1} + a_{2} i + a_{3} j + a_{4} k, i^{2} = j^{2} = k^{2} = - 1 .

And complex numbers in

C^{2}

as:

(a_{1}, a_{2}) = a_{1} + a_{2} i, i^{2} = - 1 .

In algebra B, Bermejo proposes the presence of the representation [1]:

(a_{1}, a_{2}, a_{3}) = a_{1} + a_{2} i + a_{3} j,

though he does not define it strictly, i.e., from an algebra isomorphism.

4. Defining Treons

Motivated by investigating this equivalent representation, we seek to demonstrate that this representation is mathematically correct. This implies that there must exist an isomorphism between the structure

(a_{1}, a_{2}, a_{3})

and

a_{1} + a_{2} i + a_{3} j

; both referenced by Bermejo in his definition and analysis of algebra B[1]. We assume that the field over which the algebra is defined is the real field

R

. Accordingly, we will seek an isomorphism between the real field

R^{3}

and the field of complex entities with structure

a_{1} + a_{2} i + a_{3} j

which we will call "treons" or "treonic numbers" to differentiate from the term "trions" used in various disciplines and from hypercomplex numbers called "ternions".

We define treons as:

a_{1} + a_{2} i + a_{3} j,

such that

a_{1}, a_{2}, a_{3} \in R

and

i^{2} = j^{2} = - 1

We assumed that the treonic elements

a_{1} + a_{2} i + a_{3} j

are elements of an arbitrary algebra A.

5. Addition and Product in Treons

The addition + in A is:

(a_{1} + a_{2} i + a_{3} j) + (b_{1} + b_{2} i + b_{3} j) = (a_{1} + b_{1}) + (a_{2} + b_{2}) i + (a_{3} + b_{3}) j,

and the product ⊗ in A is:

(a_{1} + a_{2} i + a_{3} j) \otimes (b_{1} + b_{2} i + b_{3} j) = (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}) + (a_{1} b_{2} + a_{2} b_{1}) i + (a_{1} b_{3} + a_{3} b_{1}) j + a_{2} i b_{3} j + a_{3} j b_{2} i .

Note that we have not imposed a definition, but have simply grouped the terms in the addition + by factoring out i and j without altering their action on the right on the elements of the field. We have also considered the distributive property of the product ⊗ over +.

6. Isomorphism with $R^{3}$

Now, we analyze

R^{3}

: The addition + in algebra B is defined as:

(a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3}) \equiv (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}) .

While the product ⊙ in B was previously defined as:

(a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) \equiv (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}) .

Under this product operation, an expression of the type

a_{3} j b_{2} i

can be expressed as [1]:

\begin{matrix} a_{3} j ⊙ b_{2} i & = a_{3} b_{2} j ⊙ i \\ = (a_{3} b_{2}) ((0, 0, 1) ⊙ (0, 1, 0)) \\ = a_{3} b_{2} (0, 1, 1) \\ = (0, a_{3} b_{2}, a_{3} b_{2}) \\ = a_{3} b_{2} (0, 1, 0) + a_{3} b_{2} (0, 0, 1) \\ = a_{3} b_{2} i + a_{3} b_{2} j . \end{matrix}

Since in algebra B[1] we have the property of orthomulearity,

a i ⊙ b j = 0

, expressions of the type

a_{2} i b_{3} j

would remain:

a_{2} i b_{3} j = a_{2} b_{3} i ⊙ j = 0 .

Taking into account that under the product in B,

a_{3} j b_{2} i = a_{3} b_{2} i + a_{3} b_{2} j

and

a_{2} i b_{3} j = 0

, then,

(a_{1} + a_{2} i + a_{3} j) \otimes (b_{1} + b_{2} i + b_{3} j) = (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}) + (a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}) i + (a_{1} b_{3} + a_{3} b_{1} + a_{3} b_{2}) j .

This has a preserved structure with respect to the product in B:

(a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) = (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}) .

Thus, for

(c_{1}, c_{2}, c_{3}) = c_{1} + c_{2} i + c_{3} j

, such that

(c_{1}, c_{2}, c_{3}) \in R^{3}

and

(c_{1} + c_{2} i + c_{3} j) \in A

, we can consider:

c_{1} = Re (c), c_{2} = {Im}_{1} (c), c_{3} = {Im}_{2} (c),

where

Re (c)

is the real part of c,

{Im}_{1} (c)

is the first imaginary part of c, and

{Im}_{2} (c)

is the second imaginary part of c.

With this, we have sufficient data to define an isomorphism

Φ

such that:

Φ : A \to R^{3},

Φ (a_{1} + a_{2} i + a_{3} j) = (a_{1}, a_{2}, a_{3}) .

7. Verification of Isomorphism Properties

7.1. Preservation of Addition

Φ ((a_{1} + a_{2} i + a_{3} j) + (b_{1} + b_{2} i + b_{3} j)) =

Φ (a_{1} + b_{1} + (a_{2} + b_{2}) i + (a_{3} + b_{3}) j) = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}) .

7.2. Preservation of Product

Φ ((a_{1} + a_{2} i + a_{3} j) \otimes (b_{1} + b_{2} i + b_{3} j)) =

Φ ((a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}) + (a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}) i + (a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}) j) =

(a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}) .

7.3. Isomorphism Verification

7.3.1. Morphism Verification

Φ

is an algebra morphism (homomorphism) if [7,8,9]:

1) Φ : A \to B,

Φ ((a_{1} + a_{2} i + a_{3} j) \otimes (b_{1} + b_{2} i + b_{3} j)) = Φ ((a_{1}, a_{2}, a_{3}) ⊙ Φ (b_{1}, b_{2}, b_{3})),

and moreover:

2) Φ {id}_{A} = {id}_{B},

where

{id}_{A}

is the identity under the product ⊗ in A, and

{id}_{B}

is the identity under the product ⊙ in B.

The first condition holds:

Φ ((a_{1} + a_{2} i + a_{3} j) \otimes (b_{1} + b_{2} i + b_{3} j)) = (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{1} + a_{3} b_{2}),

Φ (a_{1}, a_{2}, a_{3}) ⊙ Φ (b_{1}, b_{2}, b_{3}) = (a_{1} b_{1} - a_{2} b_{2} - a_{3} b_{3}, a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{2}, a_{1} b_{3} + a_{3} b_{2} + a_{3} b_{1}),

where we used

Φ (a_{k}) Φ (b_{l}) = (a_{k} + 0 i + 0 j) (b_{l} + 0 i + 0 j) = a_{k} b_{l}

The second condition also holds:

Φ {id}_{A} = {id}_{B},

The identity in A is defined as

{id}_{A} \equiv (1 + 0 i + 0 j)

(a_{1} + a_{2} i + a_{3} j) \otimes (1 + 0 i + 0 j) = (1 + 0 i + 0 j) \otimes (a_{1} + a_{2} i + a_{3} j) = (a_{1} + a_{2} i + a_{3} j),

Therefore:

Φ (1 + 0 i + 0 j) = (1, 0, 0) .

Since in algebra B we have

{id}_{B} = (1, 0, 0)

, it is verified that

Φ

is an algebra morphism between A and B.

For

Φ

to be an isomorphism,

Φ

must be both a monomorphism and an epimorphism [7,8,10].

7.3.2. Monomorphism Verification (Injectivity)

Assuming

Φ (a) = Φ (b)

. Then:

Φ (a_{1} + a_{2} i + a_{3} j) = Φ (b_{1} + b_{2} i + b_{3} j) \Rightarrow (a_{1}, a_{2}, a_{3}) = (b_{1}, b_{2}, b_{3}) .

This implies that:

a_{1} = b_{1}, a_{2} = b_{2}, a_{3} = b_{3} .

Therefore:

a_{1} + a_{2} i + a_{3} j = b_{1} + b_{2} i + b_{3} j .

Thus,

Φ

is an monomorphism.

Through the kernel, we can equally verify this. We define the kernel of

Φ

as the set of elements in A that map to the identity element of addition in B [7,8,10]. In our case:

Ker (Φ) = {a \in A ∣ Φ (a) = 0} .

For

a = a_{1} + a_{2} i + a_{3} j

, we have

Φ (a) = (a_{1}, a_{2}, a_{3})

. Therefore:

Φ (a) = 0 \Rightarrow (a_{1}, a_{2}, a_{3}) = (0, 0, 0) .

This implies that:

a_{1} = 0, a_{2} = 0, a_{3} = 0 .

Therefore:

a = 0 + 0 i + 0 j = 0 .

Thus, the only element in the kernel of

Φ

is the identity element of addition in A.

The fact that

ker (Φ)

is trivial (i.e.,

ker (Φ) = {0}

) implies that the homomorphism is injective [7,8,10].

7.3.3. Epimorphism Verification (Surjectivity)

To verify that

Φ

is an epimorphism, we considered any

(a_{1}, a_{2}, a_{3}) \in R^{3}

. There exists a treon

a_{1} + a_{2} i + a_{3} j \in A

such that:

Φ (a_{1} + a_{2} i + a_{3} j) = (a_{1}, a_{2}, a_{3}) .

Thus,

Φ

is surjective, as by definition every element in

R^{3}

has a preimage in A. Thus, by definition, it is an epimorphism.

□

With all this, we have the necessary tools to tackle the search for Euler’s identity in the context of algebra B. The isomorphism of algebras allows us to conduct a well-defined analysis of algebra B in the form of treons. With these, we proceed to deduce the form of Euler’s identity in our algebra.

8. Analysis of Treons and Their Complex Conjugates

8.1. Definition of a Complex Entity

Considering Bermejo’s work on algebra B [1], a complex entity can be described as:

b \equiv b_{1} \cdot (1, 0, 0) + b_{2} \cdot (0, 1, 0) + b_{3} \cdot (0, 0, 1) = (b_{1}, b_{2}, b_{3}),

where

b_{1} = ℜ (b)

b_{2} = ℑ_{1} (b)

, and

b_{3} = ℑ_{2} (b)

ℜ (b)

is the real part of b,

ℑ_{1} (b)

is the first imaginary part of b, and

ℑ_{2} (b)

is the second imaginary part of b. It should be noted that the definition is equivalent to the one we articulated in Section 6: Isomorphism with

R^{3}

8.2. Definition of the Complex Conjugate

The complex conjugate,

b^{*}

, was obtained by changing

i \equiv (0, 1, 0)

j \equiv (0, 0, 1)

[1] to their respective additive inverses in algebra B. Thus:

\begin{matrix} b^{(*_{i})} = b_{1} \cdot (1, 0, 0) - b_{2} \cdot (0, 1, 0) + b_{3} \cdot (0, 0, 1), \\ b^{(*_{j})} = b_{1} \cdot (1, 0, 0) + b_{2} \cdot (0, 1, 0) - b_{3} \cdot (0, 0, 1), \\ b^{(*_{i, j})} = b_{1} (1, 0, 0) - b_{2} (0, 1, 0) - b_{3} (0, 0, 1), \end{matrix}

where

b^{(*_{i})}

denotes conjugation in i, and

b^{(*_{j})}

denotes conjugation in j.

8.3. Powers of a Complex Entity

We define the product

b^{2} \equiv b ⊙ b

as:

b^{2} = (b_{1}, b_{2}, b_{3}) ⊙ (b_{1}, b_{2}, b_{3}) = (b_{1} \cdot b_{1} - b_{2} \cdot b_{2} - b_{3} \cdot b_{3}, b_{1} \cdot b_{2} + b_{2} \cdot b_{1} + b_{3} \cdot b_{2}, b_{1} \cdot b_{3} + b_{3} \cdot b_{2} + b_{3} \cdot b_{1}) .

Hence, simplifying the notation:

b_{1}^{2} = b_{1} \cdot b_{1} \equiv b_{1} b_{1} .

Thus,

b^{2}

becomes:

b^{2} = (b_{1}^{2} - b_{2}^{2} - b_{3}^{2}, 2 b_{1} b_{2} + b_{2} b_{3}, 2 b_{1} b_{3} + b_{3} b_{2}),

where we use the notation

2 b

to denote

b + b

We define higher powers as follows:

\begin{matrix} b^{3} & \equiv (b ⊙ b) ⊙ b = (b_{1}^{2} - b_{2}^{2} - b_{3}^{2}, 2 b_{1} b_{2} + b_{2} b_{3}, 2 b_{1} b_{3} + b_{3} b_{2}) ⊙ b, \\ b^{4} & \equiv ((b ⊙ b) ⊙ b) ⊙ b, \\ b^{5} & \equiv (((b ⊙ b) ⊙ b) ⊙ b) ⊙ b, \end{matrix}

and so on.

We also define the quantity

〈 b^{2} 〉 \equiv b ⊙ b^{(*_{i, j})}

. Therefore:

〈 b^{2} 〉 = (b_{1}, b_{2}, b_{3}) ⊙ {(b_{1}, b_{2}, b_{3})}^{(*_{i, j})} = {(| b |}^{2}, 2 b_{1} b_{2} + b_{2} b_{3}, 2 b_{1} b_{3} + b_{3} b_{2}),

where

{| b |}^{2} \equiv b_{1}^{2} + b_{2}^{2} + b_{3}^{2}

, which we call the "squared norm" of the vector. Note that

{| b |}^{2}

results from the definition of the product of the vector b with its complex conjugate

(*_{i, j})

, as in the field of complex numbers [11]. However, in algebra B, this quantity appears in the first component (the real component) of the vector.

9. Derivation of Euler’s Identity

9.1. Step 1: Power Series

We perform the following power series expansion [12]:

\sum_{0}^{\infty} \frac{b^{n}}{n!} \equiv \frac{b^{0}}{0!} + \frac{b^{1}}{1!} + \frac{b^{2}}{2!} + \frac{b^{3}}{3!} + \dots + \frac{b^{n}}{n!} .

where ! denotes some kind of "factorial" in the field F of algebra B [1]; therefore, the multiplication of the factorial corresponds to the multiplication · defined for the elements of the field. On the other hand, we have: 0 is the identity element of addition in F, 1 is the identity element of multiplication in F, 2 is the successor of 1, 3 is the successor of 2, and so on. We also define:

0! \equiv 1

1! \equiv 1

b^{0} \equiv (1, 0, 0)

, and

b^{1} \equiv b

. Assuming that the field F is the real field

R

, then

\sum_{n = 0}^{\infty} \frac{b^{n}}{n!}

represents the Taylor series expansion of

e^{b}

[12,13].

9.2. Step 2: Taylor Series for $e^{i b_{2}}$

We utilize the powers of the components of b multiplied by their corresponding imaginary units to perform a Taylor series expansion [13], considering algebra B over the real field

R

. For the case of the first imaginary component, we have:

e^{i b_{2}} = \sum_{0}^{\infty} \frac{{(i b_{2})}^{n}}{n!} = \frac{{(i b_{2})}^{0}}{0!} + \frac{{(i b_{2})}^{1}}{1!} + \frac{{(i b_{2})}^{2}}{2!} + \frac{{(i b_{2})}^{3}}{3!} + \dots + \frac{{(i b_{2})}^{n}}{n!},

= (1, 0, 0) + b_{2} (0, 1, 0) + \frac{{(b_{2} (0, 1, 0))}^{2}}{2!} + \frac{{(b_{2} (0, 1, 0))}^{3}}{3!} + \dots + \frac{{(b_{2} (0, 1, 0))}^{n}}{n!} .

Note that

{(i b_{2})}^{0} = {(b_{2} (0, 1, 0))}^{0} \equiv (1, 0, 0)

and

{(i b_{2})}^{1} = b_{2} (0, 1, 0) \equiv (i b_{2})

9.2.1. Analyzing the Powers of $(0, 1, 0)$

\begin{matrix} i^{0} & \equiv (1, 0, 0) \equiv i d_{⊙} (i d_{⊙} denotes the identity element of the product ⊙ in algebra B), \\ i^{1} & \equiv (0, 1, 0), \\ i^{2} & \equiv (0, 1, 0) ⊙ (0, 1, 0) = (- 1, 0, 0) = - i d_{⊙}, \\ i^{3} & \equiv ((0, 1, 0) ⊙ (0, 1, 0)) ⊙ (0, 1, 0) = (- 1, 0, 0) ⊙ (0, 1, 0) = (0, - 1, 0) = - i = i^{*}, \\ i^{4} & = (0, - 1, 0) ⊙ (0, 1, 0) = (- 1, 0, 0) = - i d_{⊙}, \\ i^{5} & = (- 1, 0, 0) ⊙ (0, 1, 0) = (0, - 1, 0) = i^{*}, \\ i^{6} & = (0, - 1, 0) ⊙ (0, 1, 0) = i^{4} = (- 1, 0, 0) = - i d_{⊙}, \\ i^{7} & = (- 1, 0, 0) ⊙ (0, 1, 0) = i^{5} = i^{*} . \end{matrix}

The sequence is cyclic and has the form, from the second power:

- i d_{⊙}, - i, - i d_{⊙}, - i, - i d_{⊙}, - i

. Therefore, the power series expansion becomes:

e^{i b_{2}} = \sum_{0}^{\infty} \frac{{(i b_{2})}^{n}}{n!} = i d_{⊙} + b_{2} i - i d_{⊙} (\frac{b_{2}^{2}}{2!} + \frac{b_{2}^{4}}{4!} + \frac{b_{2}^{6}}{6!} + \dots) + i^{*} (\frac{b_{2}^{3}}{3!} + \frac{b_{2}^{5}}{5!} + \dots),

= i d_{⊙} + b_{2} i - i d_{⊙} cosh (b_{2}) - i sinh (b_{2}),

where

cosh ()

and

sinh ()

denote hyperbolic cosines and sines [14], respectively.

9.2.2. Analyzing the Powers of $(0, 0, 1)$

\begin{matrix} j^{0} & \equiv (1, 0, 0) \equiv i d_{⊙}, \\ j^{1} & \equiv (0, 0, 1), \\ j^{2} & \equiv (0, 0, 1) ⊙ (0, 0, 1) = (- 1, 0, 0) = - i d_{⊙}, \\ j^{3} & \equiv ((0, 0, 1) ⊙ (0, 0, 1)) ⊙ (0, 0, 1) = (- 1, 0, 0) ⊙ (0, 0, 1) = (0, 0, - 1) = - j = j^{*}, \\ j^{4} & = (0, 0, - 1) ⊙ (0, 0, 1) = (- 1, 0, 0) = - i d_{⊙}, \\ j^{5} & = (- 1, 0, 0) ⊙ (0, 0, 1) = (0, 0, - 1) = j^{*}, \\ j^{6} & = (0, - 1, 0) ⊙ (0, 0, 1) = i^{4} = (- 1, 0, 0) = - i d_{⊙}, \\ j^{7} & = (- 1, 0, 0) ⊙ (0, 0, 1) = j^{5} = j^{*} . \end{matrix}

Thus:

e^{j b_{3}} = i d_{⊙} + b_{3} j - i d_{⊙} cosh (b_{3}) - j sinh (b_{3}) .

For the full vector, with its two imaginary components, we have:

e^{(b_{1} + b_{2} i + b_{3} j)} = e^{b_{1}} (i d_{⊙} + b_{2} i - i d_{⊙} cosh (b_{2}) - i sinh (b_{2})) (i d_{⊙} + b_{3} j - i d_{⊙} cosh (b_{3}) - j sinh (b_{3})) .

□

This demonstrates that Euler’s identity in algebra B, or equivalently in the algebra of treons, possesses a structure that preserves the connection between the exponential function and trigonometric functions. However, instead of the standard sine and cosine functions, hyperbolic sine and hyperbolic cosine functions appear.

Conclusions

We demonstrated that the treonic number system is a valid representation of algebra B and established an isomorphism between

R^{3}

and the algebra of treons. By preserving both the addition and product operations, our isomorphism confirmed the structural integrity of algebra B in this new representation.

Using the isomorphism between algebra B and treons, we have systematically extended the classical Euler’s identity to the domain of algebra B. We demonstrated how the exponential function and its associated trigonometric identities can be formulated within this broader context. The significance of this work lies in its ability to expand the applicability of Euler’s identity and the understanding of complex entities beyond traditional complex numbers and algebras, opening new avenues for exploring higher-dimensional algebraic structures with significant implications.

Furthermore, we demonstrated the emergence of hyperbolic trigonometric entities within this new Euler’s identity applied to algebra B. This finding underscores the potential of Euler’s identity to encompass a broader range of mathematical phenomena, which could lead to new theoretical developments in advanced mathematics.

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Analysis of Complex Entities in Algebra B

Abstract

Introduction

1. Products in Quaternion and C 2 Algebras

2. Product in Algebra B

3. Representations in Quaternion and C 2 Algebras

4. Defining Treons

5. Addition and Product in Treons

6. Isomorphism with R 3

7. Verification of Isomorphism Properties

7.1. Preservation of Addition

7.2. Preservation of Product

7.3. Isomorphism Verification

7.3.1. Morphism Verification

7.3.2. Monomorphism Verification (Injectivity)

7.3.3. Epimorphism Verification (Surjectivity)

8. Analysis of Treons and Their Complex Conjugates

8.1. Definition of a Complex Entity

8.2. Definition of the Complex Conjugate

8.3. Powers of a Complex Entity

9. Derivation of Euler’s Identity

9.1. Step 1: Power Series

9.2. Step 2: Taylor Series for e i b 2

9.2.1. Analyzing the Powers of ( 0 , 1 , 0 )

9.2.2. Analyzing the Powers of ( 0 , 0 , 1 )

Conclusions

References

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1. Products in Quaternion and $C^{2}$ Algebras

3. Representations in Quaternion and $C^{2}$ Algebras

6. Isomorphism with $R^{3}$

9.2. Step 2: Taylor Series for $e^{i b_{2}}$

9.2.1. Analyzing the Powers of $(0, 1, 0)$

9.2.2. Analyzing the Powers of $(0, 0, 1)$