Valuations as tropical hyperfield morphisms

Statement

Let $v:K\to\Gamma\cup\{\infty\}$ be a valuation on a field, where $\Gamma$ is a totally ordered abelian group, written additively. Give $\mathbb T_\Gamma=\Gamma\cup\{-\infty\}$ tropical multiplication $a\odot b=a+b$ and max-convention hyperaddition

a\boxplus b= \begin{cases} \{\max(a,b)\},&a\ne b,\\ \{c:c\le a\},&a=b. \end{cases}

Then

\operatorname{trop}_v:K\longrightarrow\mathbb T_\Gamma,\qquad \operatorname{trop}_v(0)=-\infty,\quad \operatorname{trop}_v(x)=-v(x)\ \ (x\ne0),

is a weak hyperfield homomorphism.

Verification

Multiplicativity follows from $-v(xy)=-v(x)-v(y)$ . If $v(x)\ne v(y)$ , then

v(x+y)=\min\{v(x),v(y)\},

so $\operatorname{trop}_v(x+y)$ is the unique maximum of the two target values. If the valuations tie, cancellation can only increase $v(x+y)$ , so its negative lies anywhere at or below their common value. In both cases,

\operatorname{trop}_v(x+y)\in \operatorname{trop}_v(x)\boxplus\operatorname{trop}_v(y).

Converse and sign convention

Conversely, a weak hyperfield homomorphism $w:K\to\mathbb T_\Gamma$ defines an additive valuation by $v(x)=-w(x)$ for $x\ne0$ and $v(0)=\infty$ . Thus the hyperfield-morphism axiom packages the ultrametric inequality.

The minus sign is forced by the house max convention. With a min tropical hyperfield, one may instead send $x$ directly to $v(x)$ . A multiplicative non-Archimedean absolute value maps directly to the nonnegative multiplicative presentation of the max tropical hyperfield.

Weak, not usually strong

For fixed $x,y$ , the source field has the singleton sum $\{x+y\}$ . When their tropical values tie, the target hyper-sum is an entire lower interval, so its image is usually a proper subset. The map is therefore weak and generally not strong.

References

Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: tropical hyperfields and non-Archimedean norms.
Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: hyperfield homomorphisms and the tropical hyperfield.