The Multisign Algebra: A Generalization of the Sign Concept

TL;DR

This paper introduces Multisign Algebra, a novel mathematical framework that generalizes the traditional sign concept, allowing for richer encoding of polarity within algebraic structures, potentially expanding applications in various mathematical and computational fields.

Contribution

It formally defines multisign numbers, their algebraic operations, and axioms, extending classical algebra to include a structured notion of sign beyond binary positive or negative.

Findings

Preserves key properties of classical algebraic structures.

Enables new behaviors and representations of polarity.

Provides a formal foundation for multisign algebraic systems.

Abstract

The classical number system encodes magnitude using a single scalar value whose sign positive or negative has remained conceptually unchanged for centuries. This work introduces Multisign Algebra, a mathematical generalization of the sign concept that extends the expressive capacity of real numbers. Instead of relying on a binary sign attached to a scalar value, Multisign Algebra assigns a structured scalar for sign, encoding a richer, internally organized notion of polarity within a single numerical object. We formally define multisign numbers, their algebraic operations, and the axioms that govern them, showing that this generalization preserves essential properties of classical algebraic structures while enabling new behaviors unavailable on the standard real line. This approach extends the notion of sign beyond $\pm$ and offers a refined way to encode polarity and directionality within algebraic systems.