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Algebra in the Stone-Cech Compactification Theory and Applications

:	Neil Hindman, Dona Strauss
:	Algebra in the Stone-Cech Compactification Theory and Applications
:	Walter de Gruyter GmbH& Co.KG
:	9783110258356
:	2
:	CHF 66.40
:

:	Mathematik
:	English

:	608
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

< doctype html public '-//w3c//dtd html 4.0 transitional//en'>< >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.

	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