Introduces and clarifies the basic theories of 12 structural concepts, offering a fundamental theory of groups, rings and other algebraic structures. Identifies essentials and describes interrelationships between particular theories. Selected classical theorems and results relevant to current research are proved rigorously within the theory of each structure. Throughout the text the reader is frequently prompted to perform integrated exercises of verification and to explore examples.
Author(s): Foldes S.
Publisher: Wiley
Year: 1994
Language: English
Pages: 361
Tags: Математика;Дискретная математика;
Cover......Page 1
Title......Page 4
Contents......Page 8
ILLUSTRATIONS......Page 12
PREFACE......Page 14
1. Elementary Constructions and Axioms......Page 18
2. Cardinal and Ordinal Numbers......Page 28
3. Intersections......Page 38
Bibliography......Page 45
1. Relations, Orders, and Zorn's Lemma......Page 48
2. Lattices and Closures......Page 60
3. Covering Relations......Page 66
4. Intersecting Convex Sets......Page 71
Bibliography......Page 73
1. Binary Operations, Homomorphisms, and Congruences......Page 76
2. Permutation Groups......Page 85
3. Integers and Cyclic Groups......Page 92
4. Alternating Groups......Page 99
Bibliography......Page 107
1. Ideals......Page 110
2. Polynomials......Page 121
3. Factorization and the Euclidean Algorithm......Page 128
Bibliography......Page 140
1. Rational and Real Numbers......Page 142
2. Galois Groups and Imaginary Roots......Page 154
Bibliography......Page 171
1. Bases......Page 174
2. Linear Maps and Equations......Page 184
3. Affine and Projective Geometry......Page 193
4. Hyperplanes in Linear Programming......Page 200
5. Time and Speed in Special Relativity......Page 204
Bibliography......Page 211
1. Trees and Median Graphs......Page 214
2. Games......Page 219
3. Chromatic Polynomials......Page 224
Bibliography......Page 227
1. Complements and Distributivity......Page 230
2. Boolean Algebra......Page 247
3. Modular and Geometric Lattices......Page 254
Bibliography......Page 259
1. Linear and Abstract Independence......Page 262
2. Minors and Tutte Polynomials......Page 269
3. Greedy Optimization Procedures......Page 279
Bibliography......Page 285
1. Filters......Page 286
2. Closure, Convergence, and Continuity......Page 289
3. Distances and Entourages......Page 298
Bibliography......Page 305
1. Homomorphisms and Congruences......Page 308
2. Algebra of Syntax......Page 312
3. Truth and Formal Proof......Page 317
Bibliography......Page 325
XII. CATEGORIES......Page 328
Bibliography......Page 341
INDEX OF DEFINITIONS......Page 344
INDEX OF NOTATION......Page 356
INDEX OF THEOREMS......Page 360
