Algebra in the Stone-Cech Compactification
Theory and Applications
1st Edition
Published on 20. April 2011
XIII, 485 pages
978-3-11-080922-0 (ISBN)
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08/1998
1st Edition
De Gruyter
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De Gruyter
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Content
- Intro
- I. Background Development
- Notation
- 1 Semigroups and Their Ideals
- 1.1 Semigroups
- 1.2 Idempotents and Subgroups
- 1.3 Powers of a Single Element
- 1.4 Ideals
- 1.5 Idempotents and Order
- 1.6 Minimal Left Ideals
- 1.7 Minimal Left Ideals with Idempotents
- Notes
- 2 Right Topological Semigroups
- 2.1 Topological Hierarchy
- 2.2 Compact Right Topological Semigroups
- 2.3 Closures and Products of Ideals
- 2.4 Semitopological Semigroups
- 2.5 Ellis' Theorem
- Notes
- 3 ßD
- 3.1 Ultrafilters
- 3.2 The Topological Space ßD
- 3.3 Stone-Cech Compactification
- 3.4 More Topology of ßD
- 3.5 Uniform Limits via Ultrafilters
- 3.6 The Cardinality of ßD
- Notes
- Closing Remarks
- 4 ßS
- 4.1 Extending the Operation to ßS
- 4.2 Commutativity in ßS
- 4.3 S*
- 4.4 K(ßS) and Its Closure
- Notes
- 5 ßS and Ramsey Theory
- 5.1 Ramsey Theory
- 5.2 Idempotents and Finite Products
- 5.3 Sums and Products in N
- 5.4 Adjacent Finite Unions
- 5.5 Compactness
- Notes
- II. Algebra of ßS
- 6 Ideals and Commutativity in ßS
- 6.1 The Semigroup H
- 6.2 Intersecting Left Ideals
- 6.3 Numbers of Idempotents and Ideals
- 6.4 Weakly Left Cancellative Semigroups
- 6.5 Semiprincipal Left Ideals
- 6.6 Principal Ideals in ßZ
- 6.7 Ideals and Density
- Notes
- 7 Groups in ßS
- 7.1 Zelenuk's Theorem
- 7.2 Semigroups Isomorphic to H
- 7.3 Free Semigroups and Free Groups in ßS
- Notes
- 8 Cancellation
- 8.1 Cancellation Involving Elements of S
- 8.2 Right Cancelable Elements in ßS
- 8.3 Right Cancellation in ßN and ßZ
- 8.4 Left Cancelable Elements in ßS
- 8.5 Compact Semigroups
- Notes
- 9 Idempotents
- 9.1 Right Maximal Idempotents
- 9.2 Topologies Defined by Idempotents
- 9.3 Chains of Idempotents
- 9.4 Identities in ßS
- Notes
- 10 Homomorphisms
- 10.1 Homomorphisms to the Circle Group
- 10.2 Homomorphisms from ßT into S*
- 10.3 Homomorphisms from T* into S*
- 10.4 Isomorphisms on Principal Ideals
- Notes
- 11 The Rudin-Keisler Order
- 11.1 Connections with Right Cancelability
- 11.2 Connections with Left Cancelability in N*
- 11.3 Further Connections with the Algebra of ßS
- 11.4 The Rudin-Frolík Order
- Notes
- 12 Ultrafilters Generated by Finite Sums
- 12.1 Martin's Axiom
- 12.2 Strongly Summable Ultrafilters - Existence
- 12.3 Strongly Summable Ultrafilters - Independence
- 12.4 Algebraic Properties
- Notes
- 13 Multiple Structures in ßS
- 13.1 Sums Equal to Products in ßZ
- 13.2 The Distributive Laws in ßZ
- 13.3 Ultrafilters on R Near 0
- 13.4 Left and Right Continuous Extensions
- Notes
- III. Combinatorial Applications
- 14 The Central Sets Theorem
- 14.1 Van der Waerden's Theorem
- 14.2 The Hales-Jewett Theorem
- 14.3 The Commutative Central Sets Theorem
- 14.4 The Noncommutative Central Sets Theorem
- 14.5 Combinatorial Characterization
- Notes
- 15 Partition Regularity of Matrices
- 15.1 Image Partition Regular Matrices
- 15.2 Kernel Partition Regular Matrices
- 15.3 Kernel Partition Regularity Over N
- 15.4 Image Partition Regularity Over N
- 15.5 Matrices with Entries from Fields
- Notes
- 16 IP, IP*, Central, and Central* Sets
- 16.1 Sets in Arbitrary Semigroups
- 16.2 IP* and Central Sets in N
- 16.3 IP* Sets in Weak Rings
- 16.4 Spectra and Iterated Spectra
- Notes
- 17 Sums and Products
- 17.1 Ultrafilters with Rich Structure
- 17.2 Pairwise Sums and Products
- 17.3 Sums of Products
- 17.4 Linear Combinations of Sums
- 17.5 Sums and Products in (0, 1)
- Notes
- 18 Multidimensional Ramsey Theory
- 18.1 Ramsey's Theorem and Generalizations
- 18.2 IP* Sets in Product Spaces
- 18.3 Spaces of Variable Words
- 18.4 Carlson's Theorem
- Notes
- IV. Connections With Other Structures
- 19 Relations With Topological Dynamics
- 19.1 Minimal Dynamical Systems
- 19.2 Enveloping Semigroups
- 19.3 Dynamically Central Sets
- 19.4 Dynamically Generated IP* Sets
- Notes
- 20 Density - Connections with Ergodic Theory
- 20.1 Upper Density and Banach Density
- 20.2 The Correspondence Principle
- 20.3 A Density Version of the Finite Sums Theorem
- Notes
- 21 Other Semigroup Compactifications
- 21.1 The LMC, WAP, AP, and sAP Compactifications
- 21.2 Right Topological Compactifications
- 21.3 Periodic Compactifications as Quotients
- 21.4 Spaces of Filters
- 21.5 Uniform Compactifications
- Notes
- Bibliography
- List of Symbols
- Index
