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Introduction to Lattices and Order 2nd Edition

5 out of 5 stars 2 customer reviews
ISBN-13: 978-0521784511
ISBN-10: 0521784514
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Editorial Reviews

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"...this second edition merits the same five stars as the first."
Mathematical Reviews


"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 importance of ordered structures is addressed here. Ordered structures have been increasingly recognized in recent years due to an explosion of interest in theoretical computer science and all areas of discrete mathematics. Exposure to elementary abstract algebra and the rotation of set theory are the only prerequisites for this text, intended primarily as a textbook. The level is suitable for advanced undergraduates and first year graduate students.For the new edition, much has been rewritten or expanded and new exercises have been added.
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Product Details

  • Paperback: 312 pages
  • Publisher: Cambridge University Press; 2 edition (May 6, 2002)
  • Language: English
  • ISBN-10: 0521784514
  • ISBN-13: 978-0521784511
  • Product Dimensions: 6 x 0.7 x 9 inches
  • Shipping Weight: 1.1 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #588,920 in Books (See Top 100 in Books)

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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.
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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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