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.
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.
In the min-plus algebra we have, for instance:
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.
The use of the tropical algebra in discrete event systems is handled in many sources, e.g
Jean-Eric Pin, Tropical semirings, Publ. Newton Inst. 11 (1998) 50–69 [hal:00113779, pdf]
Stéphane Gaubert, of INRIA Saclay (1999) [pdf]
Book collection of articles on idempotent semirings:
An original source:
See also:
and the first few pages of:
Iterated sums and iterated integrals over semirings, where the case of tropical semiring is a central, with applications (including in machine learning),
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