Algebra in the Stone-Cech Compactification

by Neil Hindman; Dona Strauss

• ISBN13: 9783110154207
• ISBN10: 311015420X
• Format: Nonspecific Binding
• Copyright: 2011-04-20
• Publisher: De Gruyter

Table of Contents

I. Background Development	1	(104)
Notation	2	(1)
1 Semigroups and Their Ideals	3	(28)
1.1 Semigroups	3	(5)
1.2 Idempotents and Subgroups	8	(3)
1.3 Powers of a Single Element	11	(1)
1.4 Ideals	12	(3)
1.5 Idempotents and Order	15	(4)
1.6 Minimal Left Ideals	19	(4)
1.7 Minimal Left Ideals with Idempotents	23	(7)
Notes	30	(1)
2 Right Topological Semigroups	31	(17)
2.1 Topological Hierarchy	31	(2)
2.2 Compact Right Topological Semigroups	33	(5)
2.3 Closures and Products of Ideals	38	(3)
2.4 Semitopological Semigroups	41	(2)
2.5 Ellis' Theorem	43	(4)
Notes	47	(1)
3 Beta D	48	(24)
3.1 Ultrafilters	48	(5)
3.2 The Topological Space Beta D	53	(2)
3.3 Stone-Cech Compactification	55	(3)
3.4 More Topology of Beta D	58	(4)
3.5 Uniform Limits via Ultrafilters	62	(4)
3.6 The Cardinality of Beta D	66	(3)
Notes	69	(1)
Closing Remarks	70	(2)
4 Beta S	72	(18)
4.1 Extending the Operation to Beta S	72	(8)
4.2 Commutativity in Beta S	80	(2)
4.3 S(*)	82	(3)
4.4 K (Beta S) and Its Closure	85	(3)
Notes	88	(2)
5 Beta S and Ramsey Theory	90	(15)
5.1 Ramsey Theory	90	(2)
5.2 Idempotents and Finite Products	92	(4)
5.3 Sums and Products in N	96	(1)
5.4 Adjacent Finite Unions	97	(4)
5.5 Compactness	101	(1)
Notes	102	(3)
II. Algebra of Beta S	105	(172)
6 Ideals and Commutativity in Beta S	107	(29)
6.1 The Semigroup H	107	(5)
6.2 Intersecting Left Ideals	112	(2)
6.3 Numbers of Idempotents and Ideals	114	(8)
6.4 Weakly Left Cancellative Semigroups	122	(3)
6.5 Semiprincipal Left Ideals	125	(6)
6.6 Principal Ideals in (Beta)Z	131	(2)
6.7 Ideals and Density	133	(2)
Notes	135	(1)
7 Groups in Beta S	136	(22)
7.1 Zelenuk's Theorem	136	(12)
7.2 Semigroups Isomorphic to H	148	(5)
7.3 Free Semigroups and Free Groups in Beta S	153	(4)
Notes	157	(1)
8 Cancellation	158	(28)
8.1 Cancellation Involving Elements of S	158	(3)
8.2 Right Cancelable Elements in Beta S	161	(9)
8.3 Right Cancellation in (Beta)N and (Beta)Z	170	(4)
8.4 Left Cancelable Elements in Beta S	174	(4)
8.5 Compact Semigroups	178	(7)
Notes	185	(1)
9 Idempotents	186	(19)
9.1 Right Maximal Idempotents	186	(8)
9.2 Topologies Defined by Idempotents	194	(5)
9.3 Chains of Idempotents	199	(3)
9.4 Identities in Beta S	202	(2)
Notes	204	(1)
10 Homomorphisms	205	(17)
10.1 Homomorphisms to the Circle Group	206	(4)
10.2 Homomorphisms from Beta T into S(*)	210	(4)
10.3 Homomorphisms from T(*) into S(*)	214	(4)
10.4 Isomorphisms on Principal Ideals	218	(3)
Notes	221	(1)
11 The Rudin-Keisler Order	222	(14)
11.1 Connections with Right Cancelability	223	(6)
11.2 Connections with Left Cancelability in (N)*	229	(2)
11.3 Further Connections with the Algebra of Beta S	231	(1)
11.4 The Rudin-Frolik Order	232	(2)
Notes	234	(2)
12 Ultrafilters Generated by Finite Sums	236	(22)
12.1 Martin's Axiom	236	(4)
12.2 Strongly Summable Ultrafilters -- Existence	240	(5)
12.3 Strongly Summable Ultrafilters -- Independence	245	(3)
12.4 Algebraic Properties	248	(8)
Notes	256	(2)
13 Multiple Structures in Beta S	258	(19)
13.1 Sums Equal to Products in (Beta)Z	258	(6)
13.2 The Distributive Laws in (Beta)Z	264	(4)
13.3 Ultrafilters on R Near O	268	(4)
13.4 Left and Right Continuous Extensions	272	(3)
Notes	275	(2)
III. Combinatorial Applications	277	(118)
14 The Central Sets Theorem	279	(17)
14.1 Van der Waerden's Theorem	279	(2)
14.2 The Hales-Jewett Theorem	281	(2)
14.3 The Commutative Central Sets Theorem	283	(3)
14.4 The Noncommutative Central Sets Theorem	286	(2)
14.5 Combinatorial Characterization	288	(6)
Notes	294	(2)
15 Partition Regularity of Matrices	296	(24)
15.1 Image Partition Regular Matrices	296	(5)
15.2 Kernel Partition Regular Matrices	301	(3)
15.3 Kernel Partition Regularity Over N	304	(4)
15.4 Image Partition Regularity Over N	308	(7)
15.5 Matrices with Entries from Fields	315	(3)
Notes	318	(2)
16 IP, IP(*), Central, and Central(*) Sets	320	(19)
16.1 Sets in Arbitrary Semigroups	320	(4)
16.2 IP(*) and Central Sets in XXX	324	(8)
16.3 IP(*) Sets in Weak Rings	332	(4)
16.4 Spectra and Iterated Spectra	336	(2)
Notes	338	(1)
17 Sums and Products	339	(30)
17.1 Ultrafilters with Rich Structure	339	(2)
17.2 Pairwise Sums and Products	341	(5)
17.3 Sums of Products	346	(8)
17.4 Linear Combinations of Sums	354	(7)
17.5 Sums and Products in (0, 1)	361	(6)
Notes	367	(2)
18 Multidimensional Ramsey Theory	369	(26)
18.1 Ramsey's Theorem and Generalizations	369	(7)
18.2 IP(*) Sets in Product Spaces	376	(5)
18.3 Spaces of Variable Words	381	(5)
18.4 Carlson's Theorem	386	(7)
Notes	393	(2)
IV. Connections With Other Structures	395	(60)
19 Relations With Topological Dynamics	397	(15)
19.1 Minimal Dynamical Systems	397	(3)
19.2 Enveloping Semigroups	400	(4)
19.3 Dynamically Central Sets	404	(3)
19.4 Dynamically Generated IP(*) Sets	407	(4)
Notes	411	(1)
20 Density -- Connections with Ergodic Theory	412	(13)
20.1 Upper Density and Banach Density	412	(5)
20.2 The Correspondence Principle	417	(2)
20.3 A Density Version of the Finite Sums Theorem	419	(5)
Notes	424	(1)
21 Other Semigroup Compactifications	425	(30)
21.1 The LMC, WAP, AP, and SAP Compactifications	425	(4)
21.2 Right Topological Compactifications	429	(3)
21.3 Periodic Compactifications as Quotients	432	(9)
21.4 Spaces of Filters	441	(4)
21.5 Uniform Compactifications	445	(8)
Notes	453	(2)
Bibliography	455	(16)
List of Symbols	471	(4)
Index	475