This book is intended to be a thorough introduction to the subject of ordered sets 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. The book has an excellent choice of topics, including a chapter on well ordering and ordinal numbers, which is not usually found in other texts. The approach is user-friendly and the presentation is lucid. There are more than 240 carefully chosen exercises. Topic coverage includes: modular, semimodular and distributive lattices, boolean algebras, representation of distributive lattices, algebraic lattices, congruence relations on lattices, free lattices, fixed-point theorems, duality theory and more. Steven Roman is the author of many successful textbooks, including Advanced Linear Algebra, 3rd Edition (Springer 2007), Field Theory, 2nd Edition (Springer 2005), and Introduction to the Mathematics of Finance (2004). 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. Front Matter....Pages 1-11 Front Matter....Pages 1-1 Partially Ordered Sets....Pages 2-26 Well-Ordered Sets....Pages 27-48 Lattices....Pages 49-93 Modular and Distributive Lattices....Pages 94-126 Boolean Algebras....Pages 127-142 The Representation of Distributive Lattices....Pages 143-149 Algebraic Lattices....Pages 150-164 Prime and Maximal Ideals; Separation Theorems....Pages 165-173 Congruence Relations on Lattices....Pages 174-201 Front Matter....Pages 202-202 Duality for Distributive Lattices: The Priestley Topology....Pages 203-231 Free Lattices....Pages 232-254 Fixed-Point Theorems....Pages 255-267 Back Matter....Pages 1-28
this Book Is Intended To Be A Thorough Introduction To The Subject Of Order And Lattices And Can Be Used For A Course At The Graduate Or Advanced Undergraduate Level Or For Independent Study. Prerequisites Consist Mostly Of A Bit Of Mathematical Maturity, Such As That Provided By A Basic Undergraduate Course In Abstract Algebra.
