Hyperfield - Knowlpedia

Definition

A hyperfield is a hyperring $F$ with $0\ne1$ such that $F\setminus\{0\}$ is an abelian group under its single-valued multiplication. Hyperaddition remains a canonical hypergroup operation and may be genuinely multivalued.

Fields as a special case

Every ordinary field is a hyperfield by making each sum singleton-valued. Conversely, a hyperfield with only singleton sums is an ordinary field. The Krasner, sign, and tropical hyperfields are genuinely multivalued examples.

What “division” means

Multiplicative division by a nonzero element is ordinary and unique. Additive subtraction is different: the solutions of $c\in a\boxplus x$ form a set controlled by reversibility. A hyperfield is therefore not a field with a nondeterministically chosen sum; the whole set $a\boxplus b$ is structural data.

Associated tract

The tract associated with $F$ retains precisely the finite hypersums that contain zero. This forgets the other values of a hyper-sum and does not identify arbitrary tracts with hyperfields.

References

Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: §2 and the standard examples.
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034.