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68R05 Computer science -- Discrete mathematics in relation to computer science -- Combinatorics

68R15 Computer science -- Discrete mathematics in relation to computer science -- Combinatorics on words

90-01 Operations research, mathematical programming -- Instructional exposition (textbooks, tutorial papers, etc.) En-ligne: Springerlink | Zentralblatt | MathSciNet

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This book treats the main concepts and theorems of finite ordered sets. It is well-organized and provides a very good survey over the applications of order theory. In detail the following chapters are contained (the headings in English translation): 1. Concepts and examples. This chapter presents the standard notions and many illustrations and applications. 2. Particular classes of ordered sets. This chapter contains ranked, semimodular and bipartite ordered sets, semilattices and lattices, linearly ordered sets and tournaments. 3. Morphisms of ordered sets. Here are discussed: Isotone maps, join- and meet-generating subsets, closure operations, and the Galois connection associated to a binary relation. 4. Chains and antichains. This section deals with Dilworth’s theorem and many related theorems (König-Egerváry, König-Hall, Ford/Fulkerson, Sperner). 5. Ordered sets and distributive lattices. Here, among others, dualities between preorders and topologies, and between ordered sets and distributive lattices are treated. 6. Codings and dimension of preorders. This section deals with embeddings and the Dushnik-Miller-dimension of ordered sets. 7. Some applications. This last and most extensive section contains a host of applications of order theory: models of preferences, aggregation of preferences, the roles of orders in cluster analysis, Galois data analysis: closure and implicational systems, orders in scheduling. This last section and the Appendix A (About algorithmic complexity) give an excellent survey on the state of the theory with respect to applications. (Zentralblatt)

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