## IdempotencyCertain nonlinear optimization problems arise in such areas as the theory of computation, pure and applied probability, and mathematical physics. These problems can be solved through linear methods, providing the usual number system is replaced with one that satisfies the idempotent law. Only recently has a systematic study of idempotency analysis emerged, triggered in part by a workshop organized by Hewlett-Packard's Basic Research Institute in the Mathematical Sciences (BRIMS), which brought together for the first time many leading researchers in the area. This volume, a record of that workshop, includes a variety of contributions, a broad introduction to idempotency, written especially for the book, and a bibliography of the subject. It is the most up-to-date survey currently available of research in this developing area of mathematics; the articles cover both practical and more theoretical considerations, making it essential reading for all workers in the area. |

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### Contents

An introduction to idempotency | 1 |

Tropical semirings | 50 |

Some automatatheoretic aspects of minmaxplus semirings | 70 |

The finite power property for rational sets of a free group | 80 |

The topological approach to the limitedness problem on distance automata | 88 |

Types and dynamics in partially additive categories | 112 |

Task resource models and max + automata | 133 |

Algebraic system analysis of timed Petri nets | 145 |

Maxpolynomials and discreteevent dynamic systems | 282 |

The Stochastic HJB equation and WKB method | 285 |

The Lagrange problem from the point of view of idempotent analysis | 303 |

A new differential equation for the dynamics of the Pareto sets | 322 |

Duality between probability and optimization | 331 |

topological aspects | 354 |

Random particle methods in max + optimization problems | 383 |

The geometry of finite dimensional pseudomodules | 392 |

Ergodic theorems for stochastic operators and discrete event networks | 171 |

Computational issues in recursive stochastic systems | 209 |

Periodic points of nonexpansive maps | 231 |

A systemtheoretic approach for discreteevent control of manufacturing systems | 242 |

Idempotent structures in the supervisory control of discrete event systems | 262 |

A general linear maxplus solution technique | 406 |

Axiomatics of thermodynamics and idempotent analysis | 416 |

The correspondence principle for idempotent calculus and some computer applications | 420 |

### Common terms and phrases

addition algebraic algorithms asymptotic Baccelli behavior Bellman Bellman equation bounded condition consider convex correspondence principle corresponding CTEGM CTPN D-class deﬁned Deﬁnition denote deterministic dioid discrete event networks discrete event systems distance automaton dynamics elements endomorphisms equation equivalent ergodic event graphs example exists ﬁnd ﬁnite ﬁnite automata ﬁrst formula free monoid given idempotent Idempotent analysis idempotent semiring inequality inﬁnite initial INRIA integral Kolokoltsov language Lemma linear system Markov Maslov Math Mathematics matrix max-plus min-max models monoid monotone morphism multiplication node nonexpansive obtain operations optimization variables Pareto sets Petri nets polynomial problem Proof Proposition pseudomodule Quadrat quantale random variables representation respect Rmin satisﬁes semigroup semilattice semimodule sequence solution space speciﬁcation stationary stochastic structure subset summable Theorem theory tion topical functions topology transform transition tropical semiring unique vector weak convergence workcell