Algebra in the Stone-CМЊech compactification : theory and applications

Cover......Page 1
Title......Page 4
Copyright......Page 5
Preface to the First Edition......Page 6
Preface to the Second Edition......Page 10
Notation......Page 12
Contents......Page 14
I Background Development......Page 20
1.1 Semigroups......Page 22
1.1.1 Partial Semigroups......Page 28
1.2 Idempotents and Subgroups......Page 29
1.3 Powers of a Single Element......Page 32
1.4 Ideals......Page 33
1.5 Idempotents and Order......Page 37
1.6 Minimal Left Ideals......Page 41
1.7 Minimal Left Ideals with Idempotents......Page 46
1.8 Notes......Page 56
2.1 Topological Hierarchy......Page 57
2.2 Compact Right Topological Semigroups......Page 59
2.3 Closures and Products of Ideals......Page 64
2.4 Semitopological and Topological Semigroups......Page 67
2.5 Ellis’ Theorem......Page 70
2.6 Notes......Page 74
3.1 Ultrafilters......Page 75
3.2 The Topological Space ßD......Page 80
3.3 Stone-Čech Compactification......Page 84
3.4 More Topology of ßD......Page 86
3.5 Uniform Limits via Ultrafilters......Page 93
3.6 The Cardinality of ßD......Page 97
3.7 Notes......Page 101
3.8 Closing Remarks......Page 102
4.1 Extending the Operation to ßS......Page 104
4.2 Commutativity in ßS......Page 114
4.3 S *......Page 116
4.4 K(ßS) and its Closure......Page 120
4.5 Notions of Size......Page 123
4.6 Notes......Page 125
5.1 Ramsey Theory......Page 127
5.2 Idempotents and Finite Products......Page 129
5.3 Sums and Products in N......Page 133
5.4 Adjacent Finite Unions......Page 136
5.5 Compactness......Page 139
5.6 Notes......Page 141
II Algebra of ßS......Page 144
6.1 The Semigroup H......Page 146
6.2 Intersecting Left Ideals......Page 154
6.3 Numbers of Idempotents and Ideals – Copies of H......Page 156
6.4 Weakly Left Cancellative Semigroups......Page 169
6.5 Semiprincipal Left Ideals and the Center of p(ßS)p......Page 175
6.6 Principal Ideals in ßZ......Page 180
6.7 Ideals and Density......Page 183
6.8 Notes......Page 185
7.1 Zelenyuk’s Theorem......Page 187
7.2 Semigroups Isomorphic to H......Page 201
7.3 Free Semigroups and Free Groups in ßS......Page 206
7.4 Discrete copies of Z......Page 211
7.5 Notes......Page 213
8.1 Cancellation Involving Elements of S......Page 215
8.2 Right Cancelable Elements in ßS......Page 218
8.3 Right Cancellation in ßN and ßZ......Page 227
8.4 Left Cancelable Elements in ßS......Page 231
8.5 Compact Semigroups Determined by Right Cancelable Elements in Countable Groups......Page 236
8.6 Notes......Page 244
9.1 Right Maximal Idempotents......Page 245
9.2 Topologies Defined by Idempotents......Page 254
9.3 Chains of Idempotents......Page 259
9.4 Identities in ßS......Page 264
9.5 Rectangular Semigroups in ßN......Page 265
9.6 Notes......Page 269
10 Homomorphisms......Page 271
10.1 Homomorphisms to the Circle Group......Page 272
10.2 Homomorphisms from ßT into S*......Page 276
10.3 Homomorphisms from T* into S*......Page 280
10.4 Isomorphisms Defined on Principal Left and Right Ideals......Page 285
10.5 Notes......Page 288
11 The Rudin-Keisler Order......Page 290
11.1 Connections with Right Cancelability......Page 291
11.2 Connections with Left Cancelability in N*......Page 297
11.3 Further Connections with the Algebra of ßS......Page 300
11.4 The Rudin-Frolík Order......Page 301
11.5 Notes......Page 303
12.1 Martin’s Axiom......Page 305
12.2 Strongly Summable Ultrafilters – Existence......Page 309
12.3 Strongly Summable Ultrafilters – Independence......Page 315
12.4 Algebraic Properties of Strongly Summable Ultrafilters......Page 319
12.5 Notes......Page 325
13.1 Sums Equal to Products in ßZ......Page 327
13.2 The Distributive Laws in ßZ......Page 334
13.3 Ultrafilters on R near 0......Page 337
13.4 The Left and Right Continuous Extensions of One Operation......Page 342
13.5 Notes......Page 346
III Combinatorial Applications......Page 348
14.1 Van der Waerden’s Theorem......Page 350
14.2 The Hales-Jewett Theorem......Page 352
14.3 The Commutative Central Sets Theorem......Page 354
14.4 The Noncommutative Central Sets Theorem......Page 361
14.5 A Combinatorial Characterization of Central Sets......Page 371
14.6 Geoarithmetic Progressions......Page 379
14.7 Notes......Page 382
15.1 Image Partition Regular Matrices......Page 383
15.2 Kernel Partition Regular Matrices......Page 390
15.3 Kernel Partition Regularity over N – Rado’s Theorem......Page 393
15.4 Image Partition Regularity over N......Page 397
15.5 Matrices with Entries from Fields......Page 408
15.6 Infinite Image Partition Regular Matrices......Page 412
15.7 Notes......Page 422
16.1 IP, IP*, Central, and Central* Sets in Arbitrary Semigroups......Page 424
16.2 IP* and Central Sets in N......Page 428
16.3 IP* Sets in Weak Rings......Page 437
16.4 Spectra and Iterated Spectra......Page 442
16.5 Notes......Page 444
17.1 Ultrafilters with Rich Additive and Multiplicative Structure......Page 446
17.2 Pairwise Sums and Products......Page 448
17.3 Sums of Products......Page 454
17.4 Linear Combinations of Sums - Infinite Partition Regular Matrices......Page 463
17.5 Sums and Products in (0,1) – Measurable and Baire Partitions......Page 471
17.6 Notes......Page 477
18.1 Ramsey’s Theorem and Generalizations......Page 479
18.2 IP* Sets in Product Spaces......Page 486
18.3 Spaces of Variable Words......Page 492
18.4 Carlson’s Theorem......Page 498
18.5 Notes......Page 505
IV Connections With Other Structures......Page 508
19.1 Minimal Dynamical Systems......Page 510
19.2 Enveloping Semigroups......Page 513
19.3 Dynamically Central Sets......Page 518
19.4 Dynamically Generated IP* Sets......Page 522
19.5 Notes......Page 526
20.1 Upper Density and Banach Density......Page 527
20.2 The Correspondence Principle......Page 532
20.3 A Density Version of the Finite Sums Theorem......Page 534
20.4 Notes......Page 540
21.1 The LMC, WAP, AP, and SAP Compactifications......Page 542
21.2 Right Topological Compactifications......Page 546
21.3 Periodic Compactifications as Quotients......Page 549
21.4 Semigroup Compactifications as Spaces of Filters......Page 559
21.5 Uniform Compactifications......Page 563
21.6 Notes......Page 572
Bibliography......Page 574
Index......Page 596