Finite Ordered Sets: Concepts, Results and Uses
Ordered sets are ubiquitous in mathematics and have significant applications in computer science, statistics, biology and the social sciences. As the first book to deal exclusively with finite ordered sets, this book will be welcomed by graduate students and researchers in all of these areas. Beginning with definitions of key concepts and fundamental results (Dilworth's and Sperner's theorem, interval and semiorders, Galois connection, duality with distributive lattices, coding and dimension theory), the authors then present applications 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 provided for some of the most difficult examples. The authors also point to further topics of ongoing research.
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algorithm antichain associated binary relation bipartite ordered set Boolean coding Boolean dimension Boolean lattice called Chapter characterization classes of ordered closure operator comparability graph computation consider convex Corollary correspondence covering relation cycle-free deﬁned Definition denoted Dilworth’s Theorem dimP direct product distributive lattice downsets dual closure duality equal equivalent Example Exercise exists family F family of subsets Figure function Galois connection Galois lattice given holds ifand implies incomparable instance integer intersection interval orders isomorphic isotone maps Item join-semilattice latter Lemma linear extensions linear orders mathematics maximal chains maximum median meet-irreducible meet-semilattice minimal element minimum number Monjardet Moore family notation notion obtained ordered pairs ordered subset particular partition problem Proof properties Proposition prove rank-set ranked ordered set representation residuated map respectively satisfies Section semilattices semimodular semiorders setX Sperner strict order symmetric topologies total preorders upper bound weak order