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 $x\otimes y = x+y$. This

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, in fact isomorphic 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.)

In his survey article, cited below, Pin uses the term for a wide range of similar idempotent semirings. For instance $\mathcal{M} = (\mathbb{N} \cup{\infty}, min, +)$ is a tropical semiring introduced by Imre Simon in 1978.

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 Saclay. They can be found here.

• Jean-Eric Pin, Tropical semirings, Hal preprint, hal-00113779, and in the next reference, pp.50-69, 1998, Publ. Newton Inst. 11.

Book collection of articles on idempotent semirings:

• J. Gunawadena (ed.), Idempotency, Cambridge University Press, 2001,

An original source:

• Imre Simon, (1978), Limited Subsets of a Free Monoid, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., Institute of Electrical and Electronics Engineers, 143–150 (doi:10.1109/SFCS.1978.21)

See also:

• Wikipedia, Tropical semiring