Tropical semiring

The min tropical semiring (or min-plus semiring or min-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 (or max-plus semiring or max-plus algebra or Arctic 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.

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.