# The tropical semiring

## Definitions

The tropical rig is a rig $(\mathbb{R}\cup \{\infty\}, \oplus,\otimes)$ with addition $x\oplus y = min(x,y)$ and multiplication $x\otimes y = x+y$.

The tropical semiring is a semiring $(\mathbb{R},\oplus,\otimes)$ with addition $x\oplus y = min(x,y)$ and multiplication $x\otimes y = x+y$.

Tropical geometry is often thought as the algebraic geometry over the tropical semiring.

## Terminology

The tropical rig is also called the min-plus algebra. There is a related rig called the max-plus algebra. (Some authors use the term ‘tropical algebra’ for the max-plus rather than the min-plus algebra. The theories, of course, run in parallel, as each is the negative of the other.)

## Elementary properties

The tropical semiring is an example of an idempotent semiring, since for all elements $x$, we have $x\oplus x=x$.

## Elementary example

$(5\oplus 6)\otimes 7 = 12$

## Applications

Apart from applications in tropical geometry, the min-plus and max-plus algebras have extensive use in providing algebraic models for simple discrete event systems related to timed Petri nets.

• The use of the tropical algebra in discrete event systems is handled in many sources. A slightly old set of notes (in French) for an introductory course by Stéphane Gaubert, of INRIA Rocquencourt. They can be found here.

• The book

J. Gunawadena (Editor) : Idempotency, Cambridge University Press, 2001,

contains many articles on idempotent semirings.