The tropical rig is a rig with addition and multiplication .
The tropical semiring is a semiring with addition and multiplication .
Tropical geometry is often thought as the algebraic geometry over the tropical semiring.
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.)
The tropical semiring is an example of an idempotent semiring, since for all elements , we have .
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.
J. Gunawadena (Editor) : Idempotency, Cambridge University Press, 2001,
contains many articles on idempotent semirings.