Gunawardena, Jeremy (ed.), Idempotency. Based on a workshop, Bristol, UK, October 3--7, 1994, Cambridge: Cambridge University Press. 1-49 (1998).

This paper is an excellent survey article about semirings with commutative and idempotent addition which have an absorbing zero and an identity. Such semirings are sometimes also called dioids.
After introducing some basic concepts and general results for (semirings and) dioids, the author discusses their main application areas in detail: automata and idempotency (Kleene’s theorem, the tropical dioid), discrete event systems (mathematical models, topical functions, periodicity, cycle times), nonlinear partial differential equations (viscosity solutions, functional analysis over dioids), optimisation and large deviations (idempotent measures and integration, topical functions and asymptotics).
In each section the basic ideas are explained and main results (some of them with proofs) and open problems are stated. A detailed reference list of more than 100 items is also contained. (Also submitted to MR). For the entire collection see [

Zbl 0882.00035].

##### MSC:

16Y60 | Semirings |

35A99 | General theory of PDE |

46S99 | Nonclassical functional analysis |