Algorithmic Construction of Real Hyperfields from Minimal Axioms

TL;DR

This paper introduces an optimal algorithm for classifying finite real hyperfields with cyclic positive cones, computes their C-characteristics, and provides new examples beyond Krasner's construction.

Contribution

It presents a minimal-axiom based algorithm for classifying finite real hyperfields, implements it for orders up to 17, and identifies hyperfields not derived from Krasner quotients.

Findings

Complete classification of finite real hyperfields with cyclic positive cones up to order 15.

Identification of C-characteristics in hyperfields of order up to 17.

Discovery of new hyperfields not arising from Krasner's quotient construction.

Abstract

We study real hyperfields, focusing in particular on those that are finite with cyclic positive cones. All real hyperfields have characteristic zero, although they can still be classified using the C-characteristic, an invariant that captures essential structural information. We present an algorithm to determine all such hyperfields up to isomorphism and compute their C-characteristic. The algorithm is optimal in the sense that the set of axioms used is minimal. We develop and implement this algorithm in software, enabling a complete classification of finite real hyperfields with cyclic positive cones of order up to 15, as well as identification of the C-characteristic that occur in such hyperfields of order up to 17. Restricting attention to finite hyperfields of cyclic positive cones enables substantial simplification of the algorithm, thereby enhancing its computational efficiency and allowing for the rapid generation of hyperfields of large order. Using a criterion that allows us to determine whether a given finite real hyperfield is a Krasner quotient hyperfield, we obtain many new examples of hyperfields that do not arise from Krasner's quotient construction.