nLab
tropical semiring

The tropical semiring

The tropical semiring

Definitions

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

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

Elementary example

(56)7=12(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

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:

Last revised on September 9, 2021 at 03:44:13. See the history of this page for a list of all contributions to it.