symmetric monoidal (∞,1)-category of spectra
The tropical rig is a rig $(\mathbb{R}\cup \{\infty\}, \oplus,\otimes)$ with addition $x\oplus y = min(x,y)$ and multiplication $x\otimes y = x+y$. This
The tropical semiring is a semiring $(\mathbb{R},\oplus,\otimes)$ 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.
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.
The tropical semiring is an example of an idempotent semiring, since for all elements $x$, we have $x\oplus x=x$.
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. A slightly old set of notes (in French) for an introductory course by Stéphane Gaubert, of INRIA Saclay. They can be found here.
Jean-Eric Pin, Tropical semirings, Hal preprint, hal-00113779, and in the next reference, pp.50-69, 1998, Publ. Newton Inst. 11.
Book collection of articles on idempotent semirings:
An original source:
See also:
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