✍ Scribed by George Grätzer, B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E.T. Schmidt, S.E. Schmidt, F. Wehrung, R. Wille
No coin nor oath required. For personal study only.
"Grätzer’s 'General Lattice Theory' has become the lattice theorist’s bible. Now we have the second edition, in which the old testament is augmented by a new testament. The new testament gospel is provided by leading and acknowledged experts in their fields. This is an excellent and engaging second edition that will long remain a standard reference." --MATHEMATICAL REVIEWS
Content: I First Concepts.- 1 Two Definitions of Lattices.- 2 How to Describe Lattices.- 3 Some Algebraic Concepts.- 4 Polynomials, Identities, and Inequalities.- 5 Free Lattices.- 6 Special Elements.- Further Topics and References.- Problems.- II Distributive Lattices.- 1 Characterization and Representation Theorems.- 2 Polynomials and Freeness.- 3 Congruence Relations.- 4 Boolean Algebras.- 5 Topological Representation.- 6 Pseudocomplementation.- Further Topics and References.- Problems.- III Congruences and Ideals.- 1 Weak Projectivity and Congruences.- 2 Distributive, Standard, and Neutral Elements.- 3 Distributive, Standard, and Neutral Ideals.- 4 Structure Theorems.- Further Topics and References.- Problems.- IV Modular and Semimodular Lattices.- 1 Modular Lattices.- 2 Semimodular Lattices.- 3 Geometric Lattices.- 4 Partition Lattices.- 5 Complemented Modular Lattices.- Further Topics and References.- Problems.- V Varieties of Lattices.- 1 Characterizations of Varieties.- 2 The Lattice of Varieties of Lattices.- 3 Finding Equational Bases.- 4 The Amalgamation Property.- Further Topics and References.- Problems.- VI Free Products.- 1 Free Products of Lattices.- 2 The Structure of Free Lattices.- 3 Reduced Free Products.- 4 Hopfian Lattices.- Further Topics and References.- Problems.- Concluding Remarks.- Table of Notation.- A Retrospective.- 1 Major Advances.- 2 Notes on Chapter I.- 3 Notes on Chapter II.- 4 Notes on Chapter III.- 5 Notes on Chapter IV.- 6 Notes on Chapter V.- 7 Notes on Chapter VI.- 8 Lattices and Universal Algebras.- B Distributive Lattices and Duality by B. Davey, II. Priestley.- 1 Introduction.- 2 Basic Duality.- 3 Distributive Lattices with Additional Operations.- 4 Distributive Lattices with V-preserving Operators, and Beyond.- 5 The Natural Perspective.- 6 Congruence Properties.- 7 Freeness, Coproducts, and Injectivity.- C Congruence Lattices by G. Gratzer, E. T. Schmidt.- 1 The Finite Case.- 2 The General Case.- 3 Complete Congruences.- D Continuous Geometry by F. Wehrung.- 1 The von Neumann Coordinatization Theorem.- 2 Continuous Geometries and Related Topics.- E Projective Lattice Geometries by M. Greferath, S. Schmidt.- 1 Background.- 2 A Unified Approach to Lattice Geometry.- 3 Residuated Maps.- F Varieties of Lattices by P. Jipsen, H. Rose.- 1 The Lattice A.- 2 Generating Sets of Varieties.- 3 Equational Bases.- 4 Amalgamation and Absolute Retracts.- 5 Congruence Varieties.- G Free Lattices by R. Frecse.- 1 Whitman's Solutions
Basic Results.- 2 Classical Results.- 3 Covers in Free Lattices.- 4 Semisingular Elements and Tschantz's Theorem.- 5 Applications and Related Areas.- H Formal Concept Analysis by B. Cantor and R. Wille.- 1 Formal Contexts and Concept Lattices.- 2 Applications.- 3 Sublattices and Quotient Lattices.- 4 Subdirect Products and Tensor Products.- 5 Lattice Properties.- New Bibliography.
Verbandstheorie
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Combines the techniques of an introductory text with those of a monograph to introduce the general reader to lattice theory & to bring the expert up to date on the most recent developments. This present edition has been significantly updated & expanded.
In this present edition, the work has been significantly updated and expanded. It contains an extensive new bibliography of 530 items and has been supplemented by eight appendices authored by an exceptional group of experts. The first appendix, written by the author, briefly reviews developments in
In the first half of the nineteenth century, George Boole's attempt to formalize propositional logic led to the concept of Boolean algebras. While investigating the axiomatics of Boolean algebras at the end of the nineteenth century, Charles S. Peirce and Ernst Schröder found it useful to introduce