Tropical semifield - Knowlpedia

Example

The tropical semifield in the house max-plus convention is

\mathbb T_{\max}=\mathbb R\cup\{-\infty\},\qquad x\oplus y=\max\{x,y\},\qquad x\odot y=x+y.

Its additive zero is $-\infty$ , its multiplicative one is the real number $0$ , and the multiplicative inverse of a finite $x$ is $-x$ . Hence it is an idempotent semifield.

Natural order

The natural order induced by $\oplus$ agrees with the usual order on the extended real line:

x\le_{\mathbb T}y\quad\Longleftrightarrow\quad x\oplus y=y.

Thus tropical addition is supremum and tropical multiplication is translation by ordinary addition.

Min-plus convention

The min-plus presentation uses $\mathbb R\cup\{+\infty\}$ , minimum as addition, and ordinary addition as multiplication. Negation of finite elements, together with $-\infty\leftrightarrow+\infty$ , is a semiring isomorphism from max-plus to min-plus. It reverses the usual numerical order: the natural order of the min-plus semiring is opposite to the displayed ordinary order.

Not the tropical hyperfield

The tropical hyperfield uses the same max-plus carrier and multiplication and agrees with max when two inputs differ. At a tie, however, its sum is the entire lower interval rather than a singleton. The two structures must not be identified.

References

Oleg Viro, “Hyperfields for Tropical Geometry I: Hyperfields and dequantization,” 2010. arXiv:1006.3034. Relevant: tropical algebra and tropical hyperaddition.
Grigori L. Litvinov, Viktor P. Maslov, and Grigori B. Shpiz, “Idempotent (Asymptotic) Mathematics and the Representation Theory,” 2002. arXiv:math/0206025. Relevant: max-plus idempotent mathematics.