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. This book covers areas such as ordered sets and lattices. A key feature of ordered sets, one which is emphasized in the text, is that they can be represented pictorially. Lattices are also considered as algebraic structures and hence a purely algebraic study is used to reinforce the ideas of homomorphisms and of ideals encountered in group theory and ring theory. Exposure to elementary abstract algebra and the rotation of set theory are the only prerequisites for this text. For the new edition, much has been rewritten or expanded and new exercises have been added.
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Editorial Reviews
Review
"...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
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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14 of 16 people found the following review helpful
By Ignacio
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.
19 of 32 people found the following review helpful
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.
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.
Citations (learn more)
This book cites 17
books:
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100
books
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- Stone Spaces (Cambridge Studies in Advanced Mathematics) by Peter T. Johnstone in Back Matter
- Logic for Mathematicians by A. G. Hamilton in Back Matter
- Theory and Practice of Concurrency by A. Roscoe in Back Matter
- A Shorter Model Theory by Wilfrid Hodges in Back Matter
- Introduction to Algebra by Peter J. Cameron in Back Matter
See all 17 books this book cites
- Semimodular Lattices: Theory and Applications (Encyclopedia of Mathematics and its Applications) by Manfred Stern on page 1, page 187, and Back Matter
- Combinatorics: Topics, Techniques, Algorithms by Peter J. Cameron on page 337, and Back Matter
- Formale Begriffsanalyse: Mathematische Grundlagen (German Edition) by Bernhard Ganter on page 16, and Back Matter
- Conceptual Structures at Work: 12th International Conference on Conceptual Structures, ICCS 2004, Huntsville, AL, USA, July 19-23, 2004, Proceedings ... / Lecture Notes in Artificial Intelligence) by Karl Erich Wolff on page 320, and page 389
- Graph Transformations: Second International Conference, ICGT 2004, Rome, Italy, September 28 - October 1, 2004, Proceedings (Lecture Notes in Computer Science) by Hartmut Ehrig on page 350, and page 429
See all 100 books citing this book
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