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9780521784511

Introduction to Lattices and Order

by
  • ISBN13:

    9780521784511

  • ISBN10:

    0521784514

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2002-05-06
  • Publisher: Cambridge University Press

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Summary

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Table of Contents

Preface to the second edition viii
Preface to the first edition x
Ordered sets
1(32)
Ordered sets
1(4)
Examples from social science and computer science
5(5)
Diagrams: the art of drawing ordered sets
10(4)
Constructing and de-constructing ordered sets
14(6)
Down-sets and up-sets
20(3)
Maps between ordered sets
23(10)
Exercises
25(8)
Lattices and complete lattices
33(32)
Lattices as ordered sets
33(6)
Lattices as algebraic structures
39(2)
Sublattices, products and homomorphisms
41(3)
Ideals and filters
44(2)
Complete lattices and Structures
46(4)
Chain conditions and completeness
50(3)
Join-irreducible elements
53(12)
Exercises
56(9)
Formal concept analysis
65(20)
Contexts and their concepts
65(5)
The fundamental theorem of concept lattices
70(4)
From theory to practice
74(11)
Exercises
79(6)
Modular, distributive and Boolean lattices
85(27)
Lattices satisfying additional identities
85(3)
The M3-N5 Theorem
88(5)
Boolean lattices and Boolean algebras
93(3)
Boolean terms and disjunctive normal form
96(16)
Exercises
104(8)
Representation: the finite case
112(18)
Building blocks for lattices
112(2)
Finite Boolean algebras are powerset algebras
114(2)
Finite distributive lattices are down-set lattices
116(3)
Finite distributive lattices and finite ordered sets in partnership
119(11)
Exercises
124(6)
Congruences
130(15)
Introducing congruences
130(4)
Congruences and diagrams
134(3)
The lattice of congruences of a lattice
137(8)
Exercises
140(5)
Complete lattices and Galois connections
145(30)
Closure operators
145(3)
Complete lattices coming from algebra: algebraic lattices
148(7)
Galois connections
155(10)
Completions
165(10)
Exercises
169(6)
CPOs and fixpoint theorems
175(26)
CPOs
175(5)
CPOs of partial maps
180(2)
Fixpoint theorems
182(7)
Calculating with fixpoints
189(12)
Exercises
193(8)
Domains and information systems
201(27)
Domains for computing
201(3)
Domains re-modelled: information systems
204(17)
Using fixpoint theorems to solve domain equations
221(7)
Exercises
223(5)
Maximality principles
228(19)
Do maximal elements exist? - Zorn's Lemma and the Axiom of Choice
228(4)
Prime and maximal ideals
232(5)
Powerset algebras and down-set lattices revisited
237(10)
Exercises
244(3)
Representation: the general case
247(28)
Stone's representation theorem for Boolean algebras
247(5)
Meet LINDA: the Lindenbaum algebra
252(4)
Priestley's representation theorem for distributive lattices
256(5)
Distributive lattices and Priestley spaces in partnership
261(14)
Exercises
267(8)
Appendix A: a topological toolkit 275(5)
Appendix B: further reading 280(6)
Notation index 286(3)
Index 289

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The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

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