Three-Sign Cancellation Hypernumber Systems and Associator Curvature

TL;DR

This paper introduces a novel hypernumber system extending the real numbers with a third sign, exploring its algebraic structure, associativity failures, and potential connections to hyperfields and tropical geometry.

Contribution

It constructs a three-sign cancellation hypernumber system with explicit associativity defect analysis and an ambient cancellation monoid, advancing understanding of hypernumber algebra.

Findings

Explicit formula for associativity defect $oxed{ ext{2} ext{min}(a,b)}$

Construction of a strictly associative cancellation monoid $K$

Insights into nonassociative hyperaddition over the real field

Abstract

We introduce and study a three-sign cancellation hypernumber system $H$ which extends the real field by adjoining a third sign $\Lambda$. The underlying set is $H=\{0\}\cup\{+,-,\Lambda\}\times\mathbb{R}_{>0}$, with a single-valued multiplication and a hyperaddition $\boxplus$ designed to encode cancellation phenomena between positive and negative reals. The classical real line embeds as a genuine subfield $\mathbb{R}_{\mathrm{cl}}\subset H$, and all field operations agree with the usual ones on $\mathbb{R}$. The additive structure of $H$ is almost associative but not a canonical hypergroup. We give an explicit description of where associativity fails and compute, for triples of the form $(+,a),(-,b),(\Lambda,c)$, a closed formula for the associativity defect $\kappa(a,b,c)=2\min(a,b)=a+b-|a-b|$, which coincides with the loss of absolute value when adding $a$ and $-b$ in $\mathbb{R}$. To explain this behaviour, we construct an ambient ''cancellation monoid'' $(K,\oplus)$ on $\mathbb{R}\times\mathbb{R}_{\ge 0}$ which is strictly associative and records both real sums and accumulated cancellation mass. We prove that $H$ cannot be recovered from $K$ by any simple projection, and formulate an ambient reconstruction problem. In addition, scalar multiplication by real numbers (defined via the embedded copy of $\mathbb{R}$) distributes over $\boxplus$, and the sign-layer admits a canonical hypergroup envelope governing the possible signs of hypersums. The results provide a controlled example of a nonassociative hyperaddition sitting over the real field and suggest several directions for multisign generalizations and connections with hyperfields and tropical geometry.