Tropical semiring


In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum and addition replacing the usual operations of addition and multiplication, respectively.
The tropical semiring has various applications, and forms the basis of tropical geometry.

Definition

The ' is the semiring, with the operations:
The operations ⊕ and ⊗ are referred to as tropical addition and tropical multiplication respectively. The unit for ⊕ is +∞, and the unit for ⊗ is 0.
Similarly, the ' is the semiring, with operations:
The unit for ⊕ is −∞, and the unit for ⊗ is 0.
These 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.
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: