# nLab tropical semiring

The tropical semiring

### Context

#### Algebra

higher algebra

universal algebra

# The tropical semiring

## Definitions

The tropical semiring is a semiring $(\mathbb{R}\cup \{\infty\},\oplus,\otimes)$ with addition $x\oplus y = min(x,y)$ and multiplication $x\otimes y = x+y$. (This is semiring in the sense of rig, hence sometimes the tropical rig.)

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.

## Properties

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

## Examples

In the min-plus algebra we have, for instance:

\begin{aligned} (5\oplus 6)\otimes 7 & = min(5,6) + 7 \\ & = 5 + 7 \\ & = 12 \,. \end{aligned}

## 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, e.g

Book collection of articles on idempotent semirings:

• J. Gunawardena (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)

and the first few pages of:

• William Lawvere, Introduction to Linear Categories and Applications, course lecture notes (1992) [pdf, pdf]

Iterated sums and iterated integrals over semirings, where the case of tropical semiring is a central, with applications (including in machine learning),

• Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia, Tropical time series, iterated-sums signatures, and quasisymmetric functions, SIAM Journal on Applied Algebra and Geometry 6:4 (2022) arxiv:2009.08443 doi

Last revised on July 17, 2024 at 13:51:45. See the history of this page for a list of all contributions to it.