nLab
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.

References

• 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.