The min tropical semiring (or min-plus semiring or min-plus algebra) is the semiring (, , ), with the operations:
The operations and are referred to as tropical addition and tropical multiplication respectively. The identity element for is , and the identity element for is 0.
Similarly, the max tropical semiring (or max-plus semiring or max-plus algebra or Arctic semiring) is the semiring (, , ), with operations:
The identity element unit for is , and the identity element unit for 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.
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,[2] 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.
The tropical semiring operations model how valuations behave under addition and multiplication in a valued field. A real-valued field is a field equipped with a function
which satisfies the following properties for all , in :
- if and only if
- with equality if
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:
- or with the trivial valuation, for all ,
- or its extensions with the p-adic valuation, for and coprime to ,
- the field of formal Laurent series (integer powers), or the field of Puiseux series , or the field of Hahn series, with valuation returning the smallest exponent of appearing in the series.