Steven Roman (auth.)0387789006, 9780387789002, 9780387789019
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).
Table of contents :
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
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