Lattice Theory (COLLOQUIUM PUBLICATIONS (AMER MATHEMATICAL SOC))

The purpose of the third edition is threefold: to make the deeper ideas of lattice theory accessible to mathematicians generally, to portray its structure, and to indicate some of its most interesting applications. This 1996 reprint includes expanded and updated Additional References. Lattice Theory......Page 1 Preface to the Third Edition......Page 3 Table of Contents......Page 5 1. Posets; Chains......Page 7 2. Isomorphism; Duality......Page 8 3. Diagrams; Graded Posets......Page 10 4. Lattices......Page 12 5. Lattice Algebra......Page 14 6. Distributivity......Page 17 7. Modularity......Page 18 8. Semimodularity......Page 21 9. Complemented Modular Lattices......Page 22 10. Boolean Lattices; Boolean Algebras......Page 23 1. Quasi-orderings......Page 26 2. Lattice Postulates; Semilattices......Page 27 3. Morphisms and Ideals......Page 30 4. Congruence Relations......Page 32 5. Lattice Polynomials......Page 35 6. Distributivity......Page 38 7. Modularity......Page 42 8. Semimodularity and Length......Page 45 9*. Betweenness......Page 48 10. Boolean Algebras......Page 49 11*. Brouwerian Lattices......Page 51 12*. Boolean Rings......Page 53 13*. Newman Algebras......Page 55 14*. Ortholattices......Page 58 1. Cardinal Arithmetic......Page 61 2. Formal Identities......Page 62 3. Representation of Distributive Lattices......Page 64 4. Free Distributive Lattices......Page 65 5. Free Boolean Algebras......Page 67 6. Free Modular Lattice M28......Page 69 7. Free Modular Lattices Generated by Two Chains......Page 71 8. Center......Page 72 9. Distributive and Standard Elements......Page 75 10. Sectionally Complemented Lattices......Page 76 11. Schwan's Lemma; Independence......Page 79 12. Perspectivity; Kurosh-Ore Theorem......Page 80 13. Neutral Elements in Modular Lattices......Page 83 1. Introduction......Page 86 2. Modular Pairs......Page 87 3. Examples......Page 89 4. Dependence and Rank......Page 92 5. Postulates for Geometric Lattices......Page 93 6. Modular Geometric Lattices......Page 96 7. Projective Geometries......Page 98 8. Nonmodular Geometric Lattices......Page 100 9. Partition Lattices; Algebraically Closed Subfields......Page 101 10. Graphs; Width and Breadth......Page 104 11*. Polyhedral Complexes......Page 105 12. Moebius Functions......Page 107 13*. Projectivities and Collineations......Page 110 14*. Coordinatization Problem......Page 111 15. Orthocomplementation in PGn-1(D)......Page 113 1. Closure Operations......Page 117 2. Ideal Lattices......Page 119 3. Conditional Completeness; Fixpoint Theorem......Page 120 4. Topological Closure......Page 122 5. Infinite Distributivity......Page 123 6. Lattices with Unique Complements......Page 127 7. Polarities......Page 128 8. Galois Connections......Page 130 9. Completion by Cuts......Page 132 10. Complete Brouwerian Lattices......Page 134 11*. Theorem of Glivenko......Page 135 1. Algebra......Page 138 2. Subalgebras......Page 139 3. Morphisms......Page 140 4. Congruence Relations......Page 142 5. Direct and subdirect products......Page 145 6. Free Word Algebras......Page 147 7. Free Algebras......Page 149 8. Free Lattices......Page 151 9. Postulates......Page 154 10. Families of Algebras......Page 156 11. Polymorphisms; Crypto-isomorphisms......Page 159 12*. Functors and Categories......Page 161 1. Modules; Groups with Operators......Page 165 2. Quasigroups and Loops......Page 166 3. Permutable Congruence Relations......Page 167 4. Direct Decompositions......Page 169 5. Jordan-Hölder Theorems......Page 170 6. Kurosh-Ore Theorem; Remak's Principle......Page 172 7. Theorem of Ore......Page 174 8. Subgroup Lattices......Page 176 9. Subgroups of Abelian Groups......Page 178 10. Neutral Elements; Center......Page 180 11. Modular Subgroup Lattices......Page 181 12. Jordan-Dedekind Condition and Supersolvability......Page 183 1. Ascending and Descending Chain Conditions......Page 186 2. Noetherian Distributive Lattices......Page 188 3. Finitely Generated Subalgebras......Page 189 4. Finitary Closure Properties......Page 191 5. Complete Algebraic Lattices......Page 193 6. Regular Rings......Page 195 7. Zorn's Property; Hausdorff's Axiom......Page 197 8. Subdirect Decomposition Theorem......Page 199 9. Atomic Algebraic Lattices......Page 202 10. Ordinal Sums and Products......Page 204 11. Subchains of Q and R......Page 205 12*. Homogeneous Continua; Souslin Problem......Page 207 13. Well-ordering; Ordinals......Page 208 14. Axiom of Choice......Page 211 15*. Ordinal Powers......Page 213 16*. Continuum Hypothesis; Some Uncertainties......Page 214 1. Metric Spaces......Page 217 2. Topological Spaces......Page 218 3. Directed Sets and Nets......Page 219 4. Regular Open Sets......Page 222 5. T1-lattices......Page 223 6. Lattice of Topologies; Arnold's Theorem......Page 225 7. Bases and Subbases; Compactness......Page 227 8. Alexander and Tychonoff Theorems; Compactification......Page 229 9. Stone Representation Theorem......Page 232 10. Lattices of Continuous Functions......Page 233 1. Valuations; Quasi-metric Lattices......Page 236 2. Metric Lattices; Metric Completion......Page 237 3. Distributive Valuation......Page 240 4. Valuations on Modular Lattices......Page 241 5. Continuous Geometries......Page 243 6. Jordan's Decomposition......Page 245 7. Intrinsic Topology of Chains......Page 247 8*. Dense Subsets of Chains......Page 248 9. Order and Star Convergence......Page 250 10. Star Convergence in Metric Lattices......Page 251 11. Topological Lattices......Page 253 12. Interval Topology......Page 256 1. Borel Algebras......Page 260 2. Representations of Borel Algebras......Page 261 3. Standard Borel Algebras......Page 263 4. Boolean (א,א')-algebras......Page 265 5. Finite Measures; Measure Algebras......Page 266 6. Outer Measure; Regular Measure......Page 268 7*. Existence of Measures......Page 270 8. Von Neumann Lattices......Page 273 9. Perspectivity is Transitive......Page 275 10. Dimension Functions......Page 277 11. Dimension Ortholattices......Page 279 1. Boole's Isomorphism......Page 283 2. Propositional Calculus; Critique......Page 284 3. Brouwerian and Modal Logics......Page 286 4. Attributes in Classical Mechanics......Page 288 5. Classical Probability......Page 289 6. Logic of Quantum Mechanics......Page 290 1. po-groups......Page 293 2. Directed Groups......Page 295 3. Properties of l-groups......Page 298 4. Further Algebraic Properties......Page 300 5*. Lattice-ordered Loops......Page 303 6. Discrete l-groups......Page 304 7. Ordered Groups......Page 305 8*. Orderable Groups......Page 308 9. Congruence Relations; l-ideals......Page 310 10. Principal l-ideals......Page 312 11. Commutative l-groups; Units......Page 313 12*. Structure of l-groups......Page 316 13. Complete l-groups......Page 318 14. Infinite Distributivity; Closed l-ideals......Page 320 15. Theorem of Iwasawa......Page 322 1. po-groupoids......Page 325 2. Divisibility Monoids......Page 326 3. Axioms for Magnitude......Page 328 4. Examples of l-groupoids and l-monoids......Page 329 5. Residuation......Page 331 6. Elementary Applications......Page 334 7. Integral l-groupoids......Page 335 8*. Commutation Lattices......Page 338 9. Maximal and Prime Elements......Page 340 10. Abstract Ideal Theory......Page 342 11. Fundamental Theorem of Ideal Theory......Page 344 12. Frobenius l-monoids......Page 347 13. Algebra of Relations......Page 348 14. Postulates for Relation Algebras......Page 349 1. Basic Definitions......Page 353 2. l-ideals......Page 355 3. Function Lattices......Page 357 4. Simply Ordered Vector Lattices......Page 359 5. Free Vector Lattices......Page 360 6. Integrally Closed Directed Vector Spaces......Page 362 7. Dual Spaces......Page 364 8. Complete Vector Lattices......Page 366 9. Order-convergence; Components of Weak Units......Page 367 10. Stieltjes Integral Representation......Page 368 11. Bounded Linear Functions......Page 370 12. Banach Lattices......Page 372 13. Relative Uniform Convergence......Page 375 14. Uniformly Monotone Norms......Page 377 15. (L)-spaces......Page 379 16. (M)-spaces......Page 382 17. Duality Between (L)- and (M)-spaces......Page 383 1. Introduction......Page 386 2. Hilbert's Projective Pseudo-metric......Page 387 3. Perron's Theorem......Page 389 4. Primitive Nonnegative Matrices......Page 391 5. Uniformly Semiprimitive Operators......Page 393 6. Uniformly Semiprimitive Multiplicative Processes......Page 395 7. Transition Operators......Page 396 8. Ergodic Theorem......Page 398 9. Metric Transitivity; Pointwise Ergodic Theorem......Page 400 1. po-rings and l-rings......Page 403 2. Ordered Rings and Fields......Page 404 3. L-ideals; the Radical......Page 406 4. Representations; Regular l-rings......Page 407 5. Function Rings......Page 409 6. Almost f-rings......Page 411 8. Averaging Operators......Page 413 Bibliography......Page 417 Index......Page 420 Since its original publication in 1940, this book has been revised and modernized several times, most notably in 1948 (second edition) and in 1967 (third edition). The material is organized into four main parts: general notions and concepts of lattice theory (Chapters I–V), universal algebra (Chapters VI–VII), applications of lattice theory to various areas of mathematics (Chapters VIII–XII), and mathematical structures that can be developed using lattices (Chapters XIII–XVII). At the end of the book there is a list of 166 unsolved problems in lattice theory, many of which still remain open. It is excellent reading, and ... the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavor of the field. —Bulletin of the AMS