This is an outstanding book on a field of mathematics which, although very accessible and widely applicable, is not considered as fundamental as I believe it should be. The author writes clearly, providing concise proofs with sound logic, good motivation for the material, discussion of historical development of the subject, and directions for future research. Diagrams are used very well in this text. The exercises are numerous and very illuminating, and very fun! The author provides an extensive bibliography and references results throughout the text.



There is something very enticing about the way of thinking used in lattice theory, and in particular, the way Gratzer approaches the subject in this book. Like in most areas of mathematics, a given concept can be viewed in many different ways, and one can study these objects at progressively higher levels of abstraction and generality. What is most remarkable about lattice theory, however, is that these higher levels of abstraction and generality do not become overwhelmingly difficult to comprehend as they develop--something that unfortunately happens in many other areas of mathematics. In particular, I think that algebraic concepts such as congruence relations, equational varieties, and "freeness" are much easier to understand in the context of lattices than in other algebraic structures. However, this is also true of more general mathematical concepts such as the effect of weakening conditions on theorems, searching for counterexamples, finding equivalent formulations of a given condition, and studying properties preserved under maps. This is partly due to the fact that in lattices, much of what is going on can be easily drawn or visualized. For this reason, lattices provide an excellent framework for understanding many of the basic concepts that underly all areas of mathematics.



Although the topic of this book is viewed as specialized and esoteric by some mathematicians, I believe that the material it contains is quite universal. Lattices appear in virtually every area of mathematics, and they are especially useful in universal algebra and combinatorics. While this book does not directly talk much about these applications, this book would certainly enrich the knowledge and understanding of people who work in those fields. I would recommend this book to anyone who is serious about algebra or combinatorics as well, as the ways of thinking developed by reading this book and working exercises will prove invaluable in these disciplines. This book also might be useful to beginning graduate students who want to develop their general mathematical maturity in a setting which can be a lot more fun and accessible than other areas of abstract math.



As a final note, the binding on the hardcover edition is excellent. I rarely encounter books this well-bound, in a day an age when sometimes even hardback books start falling apart after moderate use.