Core Principles and Algebraic Rules of ZIDA

Introduction

The Zero-Inclusive Division Algebra (ZIDA) is a mathematical structure that extends traditional field theory by allowing a well-defined, consistent treatment of division by zero. This is achieved through the introduction of a formal inverse of zero, denoted \( \zeta := 0^{-1} \). ZIDA is constructed to remain algebraically closed under addition, multiplication, and division, while preserving generalized consistency with limits, continuity, and differentiability.

Core Principles of ZIDA

1. Zero-Inversion Axiom

\[0 \cdot \zeta = 1 \quad \text{and hence} \quad \zeta = 0^{-1}\] This axiom introduces the element \(\zeta\), defined as the multiplicative inverse of zero.

2. Zero Regularization Principle

\[\frac{a}{0} := a \cdot \zeta, \quad \forall a \in \mathbb{ZID}\] Division by zero is defined via multiplication with \(\zeta\), where \(\mathbb{ZID}\) is the set of all ZIDA-valid elements.

3. Closure Principle

The set \(\mathbb{ZID}\) is closed under addition, subtraction, multiplication, and division, including division by zero.

4. Extended Identity and Inverse Axioms

\[a \cdot a^{-1} = 1, \quad \forall a \ne 0\] \[\text{For } a = 0: \quad 0^{-1} := \zeta, \quad \zeta^{-1} := 0\] These axioms extend the concept of inverses to include zero and its defined inverse \(\zeta\).

5. ZIDA Conjugation (Optional Extension)

\[\overline{a + b\zeta} := a - b\zeta\] This defines a conjugation operation for ZIDA-complex expressions, useful in analytic extensions and dualities.

6. ZIDA Limit Consistency

\[\lim_{x \to x_0} f(x) = a \cdot \zeta, \quad \text{where } a \in \mathbb{ZID}\] Limits that approach divergence (e.g., division by zero) are regularized via the ZIDA inverse \(\zeta\).

Algebraic Rules in ZIDA

Addition and Multiplication

\[ \begin{aligned} a + b &= b + a \quad \text{(commutativity)} \\ a \cdot b &= b \cdot a \quad \text{(commutativity)} \\ (a + b) + c &= a + (b + c) \\ (a \cdot b) \cdot c &= a \cdot (b \cdot c) \\ a \cdot (b + c) &= a \cdot b + a \cdot c \end{aligned} \]

Division Rules

\[ \begin{aligned} \frac{a}{b} &= a \cdot b^{-1}, \quad b \neq 0 \\ \frac{a}{0} &= a \cdot \zeta \\ \frac{0}{a} &= 0, \quad a \neq 0 \\ \frac{0}{0} &= 0 \cdot \zeta = 1 \end{aligned} \]

Inverse and Zero Behavior

\[ \begin{aligned} 0 \cdot \zeta &= 1 \\ \zeta^{-1} &= 0 \\ \zeta \cdot a &= a \cdot \zeta \\ \zeta + a &\in \mathbb{ZID}, \quad \forall a \in \mathbb{ZID} \end{aligned} \]

Extended Number Forms

\[ \begin{aligned} a + b\zeta &\in \mathbb{ZID}, \quad \text{for } a, b \in \mathbb{R} \text{ or } \mathbb{C} \\ (a + b\zeta)(c + d\zeta) &= ac + (ad + bc)\zeta + bd\zeta^2 \\ \zeta^2 &:= \alpha + \beta\zeta, \quad \alpha, \beta \in \mathbb{R} \end{aligned} \]

Conclusion

ZIDA generalizes arithmetic by assigning a consistent value to \(\frac{1}{0}\), avoiding the undefined behavior of division by zero. It forms a closed algebraic system suitable for extensions into calculus, complex analysis, and physics. These rules will serve as the foundation for further topological and analytic structures in the ZIDA framework.