Tropical semiring

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 min tropical semiring (or min-plus semiring or min-plus algebra) is the semiring (${\displaystyle \mathbb{R}\cup \{+\mathrm{\infty }\}}$, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$), with the operations:

${\displaystyle x\oplus y=min\{x,y\},}$

${\displaystyle x\otimes y=x+y.}$

The operations ${\displaystyle \oplus }$ and ${\displaystyle \otimes }$ are referred to as tropical addition and tropical multiplication respectively. The identity element for ${\displaystyle \oplus }$ is ${\displaystyle +\mathrm{\infty }}$, and the identity element for ${\displaystyle \otimes }$ is 0.

Similarly, the max tropical semiring (or max-plus semiring or max-plus algebra or Arctic semiring) is the semiring (${\displaystyle \mathbb{R}\cup \{-\mathrm{\infty }\}}$, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$), with operations:

${\displaystyle x\oplus y=max\{x,y\},}$

${\displaystyle x\otimes y=x+y.}$

The identity element unit for ${\displaystyle \oplus }$ is ${\displaystyle -\mathrm{\infty }}$, and the identity element unit for ${\displaystyle \otimes }$ is 0.

The two semirings are isomorphic under negation ${\displaystyle x\mapsto -x}$, 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 ⁠${\displaystyle b\to \mathrm{\infty }}$⁠ (max-plus semiring) or to zero ⁠${\displaystyle b\to 0}$⁠ (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 tropical algebra, 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.