Post-Modern Algebra

Post-Modern Algebra......Page 5
CONTENTS......Page 7
PREFACE......Page 11
2. Algebra: The Central Discipline of Mathematics......Page 15
3. Sets with Structure and Sets without Structure......Page 17
3.1 Set Mappings......Page 18
3.2. Cartesian Products and Disjoint Unions......Page 21
3.3. Relations, Good Definition, and the First Isomorphism Theorem......Page 24
3.4. Exponentiation, Quantifiers, and Negation......Page 26
3.5. Ordered Sets and Induction......Page 29
4. Semigroups and Monoids......Page 31
4.1. Free Monoids and Codes......Page 34
4.2. Dynamical Systems and Cyclic Monoids......Page 36
4.3. Semilattices and Ordered Sets......Page 38
4.4. Monoids of Relations......Page 39
1. Monoid Actions......Page 43
1.1. Automata......Page 47
1.2. The Class of All Actions......Page 50
1.3. Group Actions......Page 55
1.4. Free Groups......Page 59
1.5. Free Commutative Monoids and Partitions......Page 63
2. Groups and Quasigroups......Page 65
2.1. Multiplication Groups of Quasigroups......Page 69
2.2. Divisions and Quasigroup Homomorphisms......Page 73
2.3. Restriction and Induction......Page 77
2.4. Quasigroup and Group Conjugacy Classes......Page 79
2.5. Stability and the Class Equation......Page 82
3. Symmetry......Page 86
3.1. Permutation Groups......Page 88
3.2. Symmetric and Alternating Groups......Page 92
3.3. Sylov\'s Theorem, p-Groups and Simplicity......Page 96
4. Loops, Nets and Isotopy......Page 100
4.1. Inverse Properties and Moufang Loops......Page 105
4.2. Loop Isotopes and Bol Loops......Page 110
4.3. Right Loops and Loop Transversals......Page 114
4.4. Loop Transversal Codes......Page 118
1. General Algebra and Linear Algebra......Page 125
1.1. Products and Coproducts of Abelian Groups......Page 128
1.2. Matrices......Page 132
1.3. Unital and Non-Unital Rings......Page 136
1.4. Ideals, Fields and Domains......Page 140
2. Vector Spaces and Modules......Page 143
2.1. Duality and Transposed Matrices......Page 147
2.2. Solving Linear Equations......Page 151
2.3. Bases and Free Modules......Page 158
2.4. Determinants and Dimension......Page 164
3. Commutative Algebra......Page 172
3.1. Fractions......Page 178
3.2. Factors......Page 182
3.3. Modules over Principal Ideal Domains......Page 188
3.4. Linear Dynamical Systems......Page 195
3.5. Elementary Field Theory......Page 201
1. Posets, Monoids and Categories......Page 205
1.1. Diagonalization and Large Categories......Page 210
1.2. Functors and Concrete Categories......Page 214
1.3. Commuting Diagrams, Epimorphisms and Monomorphisms......Page 219
1.4. Natural Transformations and Functor Categories......Page 223
2. Limits and Lattices......Page 227
2.1. Products and Coproducts......Page 232
2.2. Slice Categories......Page 237
2.3. Equalizers and Pullbacks......Page 242
2.4. Groups in Categories......Page 249
2.5. Limits......Page 253
3. Adjoint Functors......Page 258
3.1. Adjunctions......Page 264
3.2. Equivalence and Boolean Algebras......Page 268
3.3. Galois Connections and Galois Theory......Page 274
3.4. Continuity and Topology......Page 280
3.5. Existence of Adjoints......Page 286
3.6. Tensor Products of Modules......Page 291
1. Sets with Operations......Page 295
1.1. Operations and Types......Page 296
1.2. The Isomorphism Theorems......Page 299
1.3. Word Algebras......Page 305
1.4. Universal Geometry......Page 310
1.5. Clones and Relations......Page 314
2. Varieties......Page 317
2.1. Replication and Prevarieties......Page 318
2.2. Bicompleteness......Page 324
2.3. Satisfaction and Varieties......Page 327
2.4. Entropic Algebras and Tensor Products......Page 332
3. Algebraic Theories......Page 337
3.1. Set-Valued Functors......Page 341
3.2. Limits of Functors......Page 345
3.3. Finitary Algebraic Theories......Page 350
3.4. Theory Maps......Page 354
4. Monads......Page 360
4.1. Monads and Their Algebras......Page 362
4.2. Monadic Adjunctions......Page 366
INDEX......Page 373