Lattices and Ordered Sets

1. Front Matter

   Pages 1-11

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2. Basic Theory

   1. Front Matter

      Pages 1-1

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   2. Partially Ordered Sets

      Pages 2-26

   3. Well-Ordered Sets

      Pages 27-48

   4. Lattices

      Pages 49-93

   5. Modular and Distributive Lattices

      Pages 94-126

   6. Boolean Algebras

      Pages 127-142

   7. The Representation of Distributive Lattices

      Pages 143-149

   8. Algebraic Lattices

      Pages 150-164

   9. Prime and Maximal Ideals; Separation Theorems

      Pages 165-173

   10. Congruence Relations on Lattices

       Pages 174-201

3. Topics

   1. Front Matter

      Pages 202-202

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   2. Duality for Distributive Lattices: The Priestley Topology

      Pages 203-231

   3. Free Lattices

      Pages 232-254

   4. Fixed-Point Theorems

      Pages 255-267

4. Back Matter

   Pages 1-28

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This book is intended to be a thorough introduction to the subject of order and lattices, with an emphasis on the latter. It can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area. This is a book on pure mathematics: I do not discuss the applications of lattice theory to physics, computer science or other disciplines. Lattice theory began in the early 1890s, when Richard Dedekind wanted to know the answer to the following question: Given three subgroups EF , and G of an abelian group K, what is the largest number of distinct subgroups that can be formed using these subgroups and the operations of intersection and sum (join), as in E?FßÐE?FÑ?GßE?ÐF?GÑ and so on? In lattice-theoretic terms, this is the number of elements in the relatively free modular lattice on three generators. Dedekind [15] answered this question (the answer is #)) and wrote two papers on the subject of lattice theory, but then the subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Grätzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann.

• Abstract algebra
• Lattice
• algebra
• category theory

From the reviews:

"This text … deals with basic material in the areas of ordered sets and lattices and is aimed at advanced undergraduate or (more likely) beginning graduate students. … The choice and coverage of the material involved is excellent and the presentation is pleasantly lucid … . Throughout the text the arguments are clear and sound, and at the end of each chapter there is a good selection of exercises to whet the reader’s appetite. … this is a worthy addition to the lattice theory literature." (T. S. Blyth, Mathematical Reviews, Issue 2009 f)

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