The congruences of a finite lattice : a proof-by-picture approach

Self-contained exposition presents the major results on congruence lattices of __finite lattices__ Includes the latest findings from a pioneering researcher in the field Features the author's signature "Proof-by-Picture" method and its conversion to transparencies Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems Excellent grad text and reference Cover ......Page 1 The Congruences of a Finite Lattice - A Proof-by-Picture Approach......Page 4 ISBN 0817632247 eISBN 0817644628 ISBN-13 9780817632243......Page 5 Contents......Page 8 Preface......Page 14 Glossary of Notation......Page 20 Picture Gallery......Page 24 Acknowledgments......Page 25 Part I: A Brief Introduction to Lattices......Page 26 1.1.1 Orders......Page 27 1.1.3 Order constructions......Page 29 1.1.4 Partitions......Page 30 1.2. Lattices and semilattices......Page 33 1.2.1 Lattices......Page 32 1.2.2 Semilattices and closure systems......Page 34 1.3.1 Homomorphisms......Page 36 1.3.2 Sublattices......Page 37 1.3.3 Congruences......Page 38 2.1. Elements and lattices......Page 43 2.2. Direct and subdirect products......Page 44 2.3. Polynomials and identities......Page 47 2.4. Gluing......Page 50 2.5.1 The characterization theorems......Page 54 2.5.2 Finite distributive lattices......Page 55 2.5.3 Finite modular lattices......Page 56 3.1. Congruence spreading......Page 59 3.2. Prime intervals......Page 61 3.3. Congruence-preserving extensions and variants......Page 63 Part II: Basic Techniques......Page 69 4.1. Basic de.nitions......Page 71 4.2. Compatible vectors of elements......Page 73 4.3. Compatible vectors of congruences......Page 74 4.4. From the chopped lattice to the ideal lattice......Page 76 4.5. Sectional complementation......Page 77 5.1. The general construction......Page 81 5.2. The congruence-preserving extension property......Page 84 5.3. The distributive case......Page 86 5.4. Two interesting intervals......Page 87 6.1. The construction......Page 95 6.2. The basic property......Page 97 Part III: Representation Theorems......Page 101 7.1. The representation theorem......Page 103 7.2. Proof-by-Picture......Page 104 7.3. Computing......Page 106 7.4. Sectionally complemented lattices......Page 107 7.5. Discussion......Page 109 8.1. The results......Page 117 8.2. Proof-by-Picture for minimal construction......Page 118 8.3. The formal construction......Page 119 8.4. Proof-by-Picture for minimality......Page 121 8.5. Computing minimality......Page 123 8.6. Discussion......Page 124 9.1. The representation theorem......Page 129 9.2. Proof-by-Picture......Page 130 9.3. Construction and proof......Page 131 9.4. Discussion......Page 138 10.1. The representation theorem......Page 139 10.2. Proof-by-Picture......Page 140 10.3. Construction and proof......Page 144 10.4. Discussion......Page 149 11.2. Proof-by-Picture......Page 153 11.3. The lattice......Page 156 11.4. Formal proof......Page 161 11.5. Discussion......Page 163 Part IV: Extensions......Page 167 12.1. The extension theorem......Page 169 12.2. Proof-by-Picture......Page 170 12.3. Simple extensions......Page 172 12.4. Formal proof......Page 174 12.5. Discussion......Page 176 13.2. Proof-by-Picture......Page 177 13.3. The conduit......Page 180 13.4. The construction......Page 181 13.6. Discussion......Page 183 14.2. Proof-by-Picture......Page 185 14.3. Formal construction......Page 189 14.4. The congruences......Page 195 14.5. The isoform property......Page 196 14.6. Discussion......Page 197 15.1. Results......Page 201 15.2. Proof-by-Picture......Page 202 15.3. Formal proofs......Page 207 15.4. Discussion......Page 211 16.1.1 Bijective maps......Page 213 16.1.2 Surjective maps......Page 214 16.2. Proof-by-Picture for bijective maps......Page 215 16.3. Verification for bijective maps......Page 218 16.4. 2/3-boolean triples......Page 222 16.5. Proof-by-Picture for surjective maps......Page 228 16.6. Verification for surjective maps......Page 230 16.7. Discussion......Page 231 Part V: Two Lattices......Page 237 17.1. The results......Page 239 17.2. Proof-by-Picture......Page 241 17.3. Multi-coloring......Page 243 17.4. Formal proof......Page 244 17.5. Discussion......Page 245 18.1. The results......Page 251 18.2. Proof-by-Picture for the main result......Page 252 18.3. A very formal proof: Main result......Page 254 18.4. Proof for sectionally complemented lattices......Page 262 18.5. Proof-by-Picture for planar lattices......Page 265 18.6. Discussion......Page 266 19.1. The problem......Page 269 19.2. Three unary functions......Page 270 19.3. De.ning tensor extensions......Page 272 19.4.1 Some special elements......Page 274 19.4.2 An embedding......Page 276 19.4.3 Distributive lattices......Page 277 19.5.1 Congruence spreading......Page 278 19.5.2 Some structural observations......Page 281 19.5.3 Lifting congruences......Page 283 19.5.4 The main lemma......Page 285 19.6. The congruence isomorphism......Page 286 19.7. Discussion......Page 287 Bibliography......Page 289 Index......Page 299

The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature Proof-by-Picture method and its conversion to transparencies.

Key features:

* Includes the latest findings from a pioneering researcher in the field

* Insightful discussion of techniques to construct nice finite lattices with given congruence lattices and nice congruence-preserving extensions

* Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems

* Additional information provided by the author online at:

http://www.maths.umanitoba.ca/homepages/gratzer.html/

The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices.

The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presents the major results on congruence lattices of finite lattices featuring the author's signature "Proof-by-Picture" method and its conversion to transparencies. Key features: * Includes the latest findings from a pioneering researcher in the field * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and nearly 80 open problems * Additional information provided by the author online at: http://www.maths.umanitoba.ca/homepages/gratzer.html/ The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices The congruences of a lattice form the congruence lattice. In the past half-century, the study of congruence lattices has become a large and important field with a great number of interesting and deep results and many open problems. This self-contained exposition by one of the leading experts in lattice theory, George Gratzer, presents the major results on congruence lattices of finite lattices featuring the author's signature Proof-by-Picture method and its conversion to transparencies. The book is appropriate for a one-semester graduate course in lattice theory, yet is also designed as a practical reference for researchers studying lattices.