Algebra in the Stone-Cech Compactification (eBook)

Theory and Applications

Neil Hindman, Dona Strauss (Autoren)

eBook Download: PDF

2012 | 2nd rev. and ext. ed.
608 Seiten
De Gruyter (Verlag)
978-3-11-025835-6 (ISBN)

Infos

This book – now in its second revised and extended edition – is a self-contained exposition of the theory of compact right semigroups for discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.

Neil Hindman, Howard University, Washington, D.C., USA; Dona Strauss, University of Leeds, United Kingdom.

Autor

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

Neil Hindman, Howard University, Washington, D.C., USA; Dona Strauss, University of Leeds, United Kingdom.

Inhalt

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

Rezensionen

lt;P>"The present book is the first devoted to an extensive study of the algebraic structure of βS and the many applications thereof; it is an exciting book, written - and very well written - by two mathematicians who are eminently qualified two write it, and it is essentially self-contained, requiring only that the reader come to it with the basic concepts of first graduate courses in algebra, analysis and topology. […] I recommend this book highly; it will be very useful, both to researchers and to students. Its index, list of symbols and up-to-date bibliography are very helpful […]."
Paul Milnes, Zentralblatt MATH / 1998

"The authors present a self-contained exposition […]. The book under review is written by two mathematicians who have contributed in a decisive way to this rapidly expanding area […] and provides a unique opportunity to obtain a 'colorful' panoramic view of the subject."
Michael Tkacenko, MathSciNet / 1999

Produkt-Details

• Erscheint lt. Verlag: 23.12.2012
• Reihe/Serie: De Gruyter Textbook
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: ergodic theory • Ramsey theory • Semigroup • Semigroup Compactification • Stone-Cech Compactification • Stone-Tschechsche Kompaktifizierung, Semigroup, Ramsey theory, Topological Dynamics, Ergodic Theory • Topological Dynamics
• ISBN-10: 3-11-025835-8 / 3110258358
• ISBN-13: 978-3-11-025835-6 / 9783110258356