Valuative hyperfield

Core idea

Let $\Gamma$ be a totally ordered abelian group, written multiplicatively. Adjoin a new element $\mathbf0$ below every element of $\Gamma$ , extend multiplication by making $\mathbf0$ absorbing, and define

x\boxplus y= \begin{cases} \{\max(x,y)\},&x\ne y,\\ \{z\in\Gamma\sqcup\{\mathbf0\}:z\leq x\},&x=y. \end{cases}

Together with the group multiplication, this makes $\Gamma_{\max}=\Gamma\sqcup\{\mathbf0\}$ a valuative hyperfield.

Hyperfield structure

The hyper-additive identity is $\mathbf0$ , while the multiplicative identity is the group identity $1_\Gamma$ . Every $x\in\Gamma$ is its own additive hyperinverse because $\mathbf0\in x\boxplus x$ . Translation-invariance of the total order makes multiplication distribute over hyperaddition.

Relation to tropical hyperfields

Taking $\Gamma=(\mathbb R,+,\leq)$ in additive notation and writing the adjoined bottom as $-\infty$ gives the max-convention tropical hyperfield. Taking the trivial one-element ordered group gives the Krasner hyperfield. Thus valuative hyperfield names the whole ordered-value-group construction, not only the real-valued example.

Valuations as morphisms

A multiplicative non-Archimedean norm $K\to\Gamma_{\max}$ is a weak hyperfield homomorphism when $K$ has singleton hyper-sums. In the additive valuation and max convention, the corresponding statement is the valuation as a tropical-hyperfield morphism proposition.

This does not assert that $\Gamma_{\max}$ is an orbit quotient of every field carrying such a valuation. Quotient hyperfields retain the realized orbit sums, whereas this construction depends only on the ordered group.

References

Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707. Relevant: Example 2.12 on valuative hyperfields.
Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: tropical hyperfields and non-Archimedean norms.