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Introduction to Lattices and Order Paperback – 18 Apr 2002


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Product details

  • Paperback: 309 pages
  • Publisher: Cambridge University Press; 2 edition (18 April 2002)
  • Language: English
  • ISBN-10: 0521784514
  • ISBN-13: 978-0521793629
  • Product Dimensions: 15.2 x 1.7 x 22.8 cm
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 591,663 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Review

'Recommended for every academic mathematics collection.' Choice

'… an excellent introductory textbook on ordered sets and lattices and it is intended for undergraduate and beginning graduate students in mathematics.' Vaclav Slavic, Zentralblatt für Mathematik

'I used Introduction to Lattices and Order as the sole textbook in a one semester course. The students enrolled were a heterogeneous group including modestly prepared undergraduates, well trained graduate students, and a few applications-oriented computer science students … In short, the textbook was a success.' Joel Berman, Australian Mathematical Society Gazette

'… a well-written, satisfying, informative, and stimulating account of applications that are of great interest, particularly in computer science and social science … it will surely become a classic.' Mathematical Reviews

'Altogether, this is a great book. It would be interesting (and educational) to give a course based on it - almost makes me wish I hadn't retired!' Australian Mathematical Society Gazette

'… a valuable source to anyone who needs to use ordered structures in any context.' EMS Newsletter

'It can be recommended as a valuable source to anyone who needs to use ordered structures in any context.' European Mathematical Society

'The book is written in a very engaging and fluid style. The understanding of the content is aided tremendously by the very large number of beautiful lattice diagrams … The book provides a wonderful and accessible introduction to lattice theory, of equal interest to both computer scientists and mathematicians.' Jonathan Cohen, SIGACT News

Book Description

The explosive development of theoretical computer science in recent years has influenced this new edition: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added.

Inside This Book (Learn More)
First Sentence
Order, order, order - it permeates mathematics, and everyday life, to such an extent that we take it for granted. Read the first page
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2 of 2 people found the following review helpful By Robert on 9 Feb. 2010
Format: Paperback
Lattices and order is one of those fields like knots or graphs which has a visual appeal. Readers coming to this book thinking that might be what they are getting should be warned that the presentation takes no prisoners. There are no gentle build ins to problems, no crib for the problems, and the presentation of ideas is extremely formal with no room taken to clarify concepts. It is not a bad book but if you plan to work through it yourself or even to use it for teaching it will prove extremely challenging.
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 2 reviews
16 of 18 people found the following review helpful
Excellent introduction and something more 28 Mar. 2000
By Ignacio - Published on Amazon.com
Format: Paperback
This book presents an excellent introduction to the subject, but also goes beyond that, presenting with a fair amount of the detail the theory of Priestley representation. The excercises start at the basic level of checking the understanding of definitions, allowing the reader to build confidence out of the practice. The fact that Priestley herself co-authored it is definitely a plus.
25 of 39 people found the following review helpful
Lattice Theory Uber Alles?Begin here 27 Mar. 2005
By galloamericanus - Published on Amazon.com
Format: Paperback
A set with, at minimum, one binary operation is a groupoid. If a situation involves an equivalence relation or some sort of symmetry, some sort of groupoid applies. If the set has, at minimum, two binary operations, and one operation distributes over the other, you have a ringoid. Ringoids, which include the real field we all use every day, tell us much about number systems.

Let there be a groupoid. Denote its single binary operation by concatenation. Let that operation commute and associate. So far, we have a commutative semigroup. Now add idempotency, so that AA=A. With that seemingly trivial axiom we turn a corner, farewell the groupoids, and find ourselves among the semilattices.

Now let there be two binary operations, + and *, that commute and associate. Moreover, assume that A*(A+B) = A = A+(A*B). A*A=A=A+A is now an easy theorem. What you now have is a lattice, of which the best known example is Boolean algebra (which requires added axioms). More generally, most logics can be seen as interpretations of bounded lattices. Given any relation of partial or total order, the corresponding algebra is lattice theory. Nevertheless, far fewer mathematicians specialize in lattices than in groupoids and ringoids.

Davey and Priestley has become the classic introduction to lattice theory in our time. Sad to say, it has little competition. It is a bit harder than I would prefer, and the authors do not say enough about the value of lattice theory for nonclassical logic. Their book is a classic nonetheless, and here's hoping that Gian Carlo Rota was right when he said that the 21st century shall be the century of lattices triumphant.

Lattice theory is largely due to the work of the American Garrett Birkhoff, writing in the 1930s. He gets my vote for the

greatest American mathematician of all time.
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