Tropical semiring

The {{visible anchor|min tropical semiring}} (or {{visible anchor|min-plus semiring}} or {{visible anchor|min-plus algebra}}) is the semiring ($\backslash mathbb\{R\}\; \backslash cup\; \backslash \{+\backslash infty\backslash \}$, $\backslash oplus$, $\backslash otimes$), with the operations:

: $x\; \backslash oplus\; y\; =\; \backslash min\backslash \{x,\; y\; \backslash \},$

: $x\; \backslash otimes\; y\; =\; x\; +\; y.$

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

Similarly, the {{visible anchor|max tropical semiring}} (or {{visible anchor|max-plus semiring}} or {{visible anchor|max-plus algebra}} or {{visible anchor|Arctic semiring}}{{Citation needed|date=April 2025}}) is the semiring ($\backslash mathbb\{R\}\; \backslash cup\; \backslash \{-\backslash infty\backslash \}$, $\backslash oplus$, $\backslash otimes$), with operations:

: $x\; \backslash oplus\; y\; =\; \backslash max\backslash \{x,\; y\; \backslash \},$

: $x\; \backslash otimes\; y\; =\; x\; +\; y.$

The identity element unit for $\backslash oplus$ is $-\backslash infty$, and the identity element unit for $\backslash otimes$ is 0.

The two semirings are isomorphic under negation $x\; \backslash mapsto\; -x$, 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 {{tmath|b \to \infty}} (max-plus semiring) or to zero {{tmath|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 {{visible anchor|tropical algebra}},{{cite book|last1=Litvinov|first1=Grigoriĭ Lazarevich|last2=Sergeev|first2=Sergej Nikolaevič|title=Tropical and Idempotent Mathematics: International Workshop Tropical-07, Tropical and Idempotent Mathematics|date=2009|publisher=American Mathematical Society| isbn=9780821847824|page=8|url=http://www.mccme.ru/tropical12/Tropics2012final.pdf|accessdate=15 September 2014}} 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.

{{main|Valued field}}

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

: $v:K\; \backslash to\; \backslash R\; \backslash cup\; \backslash \{\backslash infty\backslash \}$

which satisfies the following properties for all $a$, $b$ in $K$:

: $v(a)\; =\; \backslash infty$ if and only if $a\; =\; 0,$

: $v(ab)\; =\; v(a)\; +\; v(b)\; =\; v(a)\; \backslash otimes\; v(b),$

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

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:

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

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• {{cite arXiv |author-link= |eprint=math/0507014v1 |title= The Maslov dequantization, idempotent and tropical mathematics: A brief introduction|class= |last1= Litvinov|first1= G. L.|year= 2005}}

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Semiring