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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.^[1]

Definition[edit]

The min tropical semiring (ormin-plus semiringormin-plus algebra) is the semiring (${\displaystyle \mathbb {R} \cup \{+\infty \}}$, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$), 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${\displaystyle +\infty }$, and the identity element for ${\displaystyle \otimes }$ is 0.

Similarly, the max tropical semiring (ormax-plus semiringormax-plus algebraorArctic semiring) is the semiring (${\displaystyle \mathbb {R} \cup \{-\infty \}}$, ${\displaystyle \oplus }$, ${\displaystyle \otimes }$), with operations:

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

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

The identity element unit for ${\displaystyle \oplus }$is${\displaystyle -\infty }$, 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 ${\displaystyle b\to \infty }$ (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,^[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.

Valued fields[edit]

The tropical semiring operations model how valuations behave under addition and multiplication in a valued field. A real-valued field ${\displaystyle K}$ is a field equipped with a function

${\displaystyle v:K\to \mathbb {R} \cup \{\infty \}}$

which satisfies the following properties for all ${\displaystyle a}$, ${\displaystyle b}$in${\displaystyle K}$:

${\displaystyle v($a)=\infty } if and only if ${\displaystyle a=0,}$

${\displaystyle v($ab)=v(a)+v(b)=v(a)\otimes v(b),}

${\displaystyle v(a+b)\geq \min\{v($a),v(b)\}=v(a)\oplus v(b),} with equality if ${\displaystyle v($a)\neq v(b).}

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:

• ${\displaystyle \mathbb {Q} }$or${\displaystyle \mathbb {C} }$ with the trivial valuation, ${\displaystyle v($a)=0} for all ${\displaystyle a\neq 0}$,
• ${\displaystyle \mathbb {Q} }$ or its extensions with the p-adic valuation, ${\displaystyle v(p^{n}a/b)=n}$ for ${\displaystyle a}$ and ${\displaystyle b}$ coprime to ${\displaystyle p}$,
• the field of formal Laurent series ${\displaystyle K(($t))} (integer powers), or the field of Puiseux series ${\displaystyle K\{\{t\}\}}$, or the field of Hahn series, with valuation returning the smallest exponent of ${\displaystyle t}$ appearing in the series.

References[edit]