Finite Ordered Sets Concepts, Results and Uses
Résumé
Ordered sets are ubiquitous in mathematics and have significant applications in computer science, statistics, biology and the social sciences. We present the first book to dealing exclusively with finite ordered sets. Five chapters are devoted to definitions of key concepts and fundamental results (ranked orders, Dilworth's and Sperner's theorem, Galois connection and residuation, duality between orders and distributive lattices, coding and dimension theory). The last - and larger - chapter presents uses of these structures in fields such as preference modelling and aggregation, operational research and management, cluster and concept analysis, and data mining. Exercises are included at the end of each chapter with helpful hints or references provided for the most difficult ones. We also point to further topics of ongoing research. At last there are appendices devoted to algorithmic complexity, documentation marks, types and numbers of ordered sets, about 500 references, a list of symbols and a (substantial) index.
Les ensembles ordonnés sont partout en mathématiques et ont d'importantes applications en informatique, statistique, biologie et sciences sociales. Dans ce livre -le premier consacré exclusivement aux ensembles ordonnés finis- cinq chapitres donnent les concepts de base et les résultats fondamentaux (théorèmes de Dilworth et Sperner, correspondance de Galois, dualité ordres treillis distributifs, codages et dimensions,..) tandis que le sixième présente de nombreux usages de ces structures dans des domaines variés (tels que choix individuel ou collectif, recherche opérationnelle, analyse et fouille des données, analyse des concepts...). Il y a de nombreux exercices, quatre annexes (complexité algorithmique, ressources documentaires...) et les sections compléments des chapitre citent près de 500 références.