Algebra in the Stone-Cech Compactification: Theory and Applications (De Gruyter Expositions in Mathematics, 27)

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 6
Contents\0......Page 10
I. Background Development......Page 16
Notation\0......Page 17
1.1 Semigroups\0......Page 18
1.2 Idempotents and Subgroups\0......Page 23
1.3 Powers of a Single Element\0......Page 26
1.4 Ideals\0......Page 27
1.5 Idempotents and Order\0......Page 30
1.6 Minimal Left Ideals\0......Page 34
1.7 Minimal Left Ideals with Idempotents\0......Page 38
Notes\0......Page 45
2.1 Topological Hierarchy\0......Page 46
2.2 Compact Right Topological Semigroups\0......Page 48
2.3 Closures and Products of Ideals\0......Page 53
2.4 Semitopological Semigroups\0......Page 56
2.5 Ellis\' Theorem\0......Page 58
Notes\0......Page 62
3.1 Ultrafilters\0......Page 63
3.2 The Topological Space \\beta D\0......Page 68
3.3 Stone-tech Compactification\0......Page 70
3.4 More Topology of \\beta D\0......Page 73
3.5 Uniform Limits via Ultrafilters\0......Page 77
3.6 The Cardinality of \\beta D\0......Page 81
Notes\0......Page 84
Closing Remarks\0......Page 85
4.1 Extending the Operation to S\0......Page 87
4.2 Commutativity in \\beta S\0......Page 95
4.3 S*\0......Page 97
4.4 K(\\beta S) and Its Closure\0......Page 100
Notes\0......Page 103
5.1 Ramsey Theory\0......Page 105
5.2 Idempotents and Finite Products\0......Page 107
5.3 Sums and Products in N\0......Page 111
5.4 Adjacent Finite Unions\0......Page 112
5.5 Compactness\0......Page 116
Notes\0......Page 117
II. Algebra of \\beta S\0......Page 120
6.1 The Semigroup H\0......Page 122
6.2 Intersecting Left Ideals\0......Page 127
6.3 Numbers of Idempotents and Ideals\0......Page 129
6.4 Weakly Left Cancellative Semigroups\0......Page 137
6.5 Semiprincipal Left Ideals\0......Page 140
6.6 Principal Ideals in \\beta Z\0......Page 146
6.7 Ideals and Density\0......Page 148
Notes\0......Page 150
7.1 Zelenuk\'s Theorem\0......Page 151
7.2 Semigroups Isomorphic to H\0......Page 163
7.3 Free Semigroups and Free Groups in \\beta S\0......Page 168
Notes\0......Page 172
8.1 Cancellation Involving Elements of S\0......Page 173
8.2 Right Cancelable Elements in 8S\0......Page 176
8.3 Right Cancellation in,8N and $7G\0......Page 185
8.4 Left Cancelable Elements in $S\0......Page 189
8.5 Compact Semigroups\0......Page 193
Notes\0......Page 200
9.1 Right Maximal Idempotents\0......Page 201
9.2 Topologies Defined by Idempotents\0......Page 209
9.3 Chains of Idempotents\0......Page 214
9.4 Identities in $S\0......Page 217
Notes\0......Page 219
10 Homomorphisms\0......Page 220
10.1 Homomorphisms to the Circle Group\0......Page 221
10.2 Homomorphisms from 1ST into S*\0......Page 225
10.3 Homomorphisms from T* into S*\0......Page 229
10.4 Isomorphisms on Principal Ideals\0......Page 233
Notes\0......Page 236
11 -The Rudin-Keisler Order\0......Page 237
11.1 Connections with Right Cancelability\0......Page 238
11.2 Connections with Left Cancelability in N*\0......Page 244
11.3 Further Connections with the Algebra of,8S\0......Page 246
11.4 The Rudin-Frolilc Order\0......Page 247
Notes\0......Page 249
12.1 Martin\'s Axiom\0......Page 251
12.2 Strongly Summable Ultrafilters - Existence\0......Page 255
12.3 Strongly Summable Ultrafilters - Independence\0......Page 260
12.4 Algebraic Properties\0......Page 263
Notes\0......Page 271
13.1 Sums Equal to Products in flZ\0......Page 273
13.2 The Distributive Laws in PZ\0......Page 279
13.3 Ultrafilters on R Near 0\0......Page 283
13.4 Left and Right Continuous Extensions\0......Page 287
Notes\0......Page 290
III. Combinatorial Applications\0......Page 292
14.1 Van der Waerden\'s Theorem\0......Page 294
14.2 The Hales-Jewett Theorem\0......Page 296
14.3 The Commutative Central Sets Theorem\0......Page 298
14.4 The Noncommutative Central Sets Theorem\0......Page 301
14.5 Combinatorial Characterization\0......Page 303
Notes\0......Page 309
15.1 Image Partition Regular Matrices\0......Page 311
15.2 Kernel Partition Regular Matrices\0......Page 316
15.3 Kernel Partition Regularity Over N\0......Page 319
15.4 Image Partition Regularity Over N\0......Page 323
15.5 Matrices with Entries from Fields\0......Page 330
Notes\0......Page 333
16.1 Sets in Arbitrary Semigroups\0......Page 335
16.2 IP* and Central Sets in N\0......Page 339
16.3 IP* Sets in Weak Rings\0......Page 347
16.4 Spectra and Iterated Spectra\0......Page 351
Notes\0......Page 353
17.1 Ultrafilters with Rich Structure\0......Page 354
17.2 Pairwise Sums and Products\0......Page 356
17.3 Sums of Products\0......Page 361
17.4 Linear Combinations of Sums\0......Page 369
17.5 Sums and Products in (0, 1)\0......Page 376
Notes\0......Page 382
18.1 Ramsey\'s Theorem and Generalizations\0......Page 384
18.2 IP* Sets in Product Spaces\0......Page 391
18.3 Spaces of Variable Words\0......Page 396
18.4 Carlson\'s Theorem\0......Page 401
Notes\0......Page 408
IV. Connections With Other Structures\0......Page 410
19.1 Minimal Dynamical Systems\0......Page 412
19.2 Enveloping Semigroups\0......Page 415
19.3 Dynamically Central Sets\0......Page 419
19.4 Dynamically Generated IP* Sets\0......Page 422
Notes\0......Page 426
20.1 Upper Density and Banach Density\0......Page 427
20.2 The Correspondence Principle\0......Page 432
20.3 A Density Version of the Finite Sums Theorem\0......Page 434
Notes\0......Page 439
21.1 The L.MC, W,4JP, AY, and -SA>Compactifications\0......Page 440
21.2 Right Topological Compactifications\0......Page 444
21.3 Periodic Compactifications as Quotients\0......Page 447
21.4 Spaces of Filters\0......Page 456
21.5 Uniform Compactifications\0......Page 460
Notes\0......Page 468
Bibliography......Page 470
List of Symbols......Page 486
Index......Page 490