Knowlpedia

Example

Tropical hyperfield

The max-plus hyperfield whose tied sum is the full lower interval.

The tropical hyperfield in the house max convention has carrier

T=R∪{−∞}.\mathbb T=\mathbb R\cup\{-\infty\}.

Its multiplication is x⊙y=x+yx\odot y=x+y, with multiplicative identity 00 and absorbing element −∞-\infty. Its hyperaddition is

x⊞y={{max⁡(x,y)},x≠y,{z∈T:z≤x},x=y.x\boxplus y= \begin{cases} \{\max(x,y)\},&x\ne y,\\ \{z\in\mathbb T:z\le x\},&x=y. \end{cases}

The additive identity is −∞-\infty, and every finite element is its own additive inverse because −∞∈x⊞x-\infty\in x\boxplus x.

The tied-sum mechanism

When one input is strictly larger, it uniquely dominates the hyper-sum. When the inputs tie, cancellation can lower the result by any amount, even to the additive zero. This is the hyperfield form of the non-Archimedean rule that equal leading valuations may cancel.

Other presentations

Exponentiating finite elements gives the equivalent carrier R≥0\mathbb R_{\ge0}, with ordinary multiplication and

a⊞b={{max⁡(a,b)},a≠b,[0,a],a=b.a\boxplus b= \begin{cases} \{\max(a,b)\},&a\ne b,\\ [0,a],&a=b. \end{cases}

Some sources instead use a min convention. The max convention here is chosen to align weak morphisms K→TK\to\mathbb T with non-Archimedean absolute values, or with the negative of an .

References
  1. Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: tropical real hyperfields and dequantization.
  2. Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: Example 2.5.