Introduction to Lattices and Order
This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.
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apply asserts assume Boolean algebra called Chapter claim clopen closed closure operator compact complete lattice concept condition congruence Consider consistent construction contains context continuous corresponding defined definition denoted diagram directed distributive lattice domain down-set Draw dual duality easy element equivalent example Exercise exists fact Figure filter final finite formal function Further Galois connection given gives Hence holds homomorphism ideal implies important inclusion infinite information systems isomorphic join join-irreducible label least fixpoint Lemma maximal element meet models modular n-structure natural non-empty Note numbers objects obtained order-isomorphism ordered set partial particular presented preserves prime ideal proof properties Proposition Prove provides relation representation result satisfies Show space strings structure sublattice subset Table term Theorem theory topology topped union upper bound write