In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name tropical is a reference to the Hungarian-born computer scientist Imre Simon, so named because he lived and worked in Brazil.

Definition

The (or ' or ') is the semiring (, \oplus, \otimes), with the operations: The operations \oplus and \otimes are referred to as tropical addition and tropical multiplication respectively. The identity element for \oplus is +\infty, and the identity element for \otimes is 0. Similarly, the ' (or ' or ' or **''') is the semiring (, \oplus, \otimes), with operations: The identity element unit for \oplus is -\infty, and the identity element unit for \otimes is 0. The two semirings are isomorphic under negation, and generally one of these is chosen and referred to simply as the tropical semiring. Conventions differ between authors and subfields: some use the min convention, some use the max convention. The two tropical semirings are the limit ("tropicalization", "dequantization") of the log semiring as the base goes to infinity b \to \infty (max-plus semiring) or to zero b \to 0 (min-plus semiring). Tropical addition is idempotent, thus a tropical semiring is an example of an idempotent semiring. A tropical semiring is** also**** referred**** to**** as**** a , though**** this**** should**** not be**** confused**** with**** an**** associative algebra over**** a tropical**** semiring****.**** Tropical exponentiation is defined in the usual way as iterated tropical products.

Valued fields

The tropical semiring operations model how valuations behave under addition and multiplication in a valued field. A real-valued field K is a field equipped with a function which satisfies the following properties for all a, b in K: Therefore the valuation v is almost a semiring homomorphism from K to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together. Some common valued fields: