The Multisign Algebra: A Generalization of the Sign Concept

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.

arXiv comment28 pages, Second version. Revised proofs and clarified definitions. The definition of precedence was removed. Several remarks were added to clarify points that were unclear in the previous version. Definitions were made more explicit throughout the manuscript. Errors in some proofs were corrected. The appendices were revised accordingly