nLab
tropical semiring

Definitions

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

The tropical semiring is a semiring $(ℝ,\oplus ,\otimes )$ 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.)

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 system?s related to timed Petri nets.

References

• The use of the tropical algebra in discrete event system?s 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.