Algebra in the Stone-Cech Compactification: Theory and Applications

Description

This is the second revised and extended edition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition. This book 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. Second revised and extended edition, now in paperback With lots of exercises Includes new results obtained during the past thirteen years Preface to the First Edition Preface to the Second Edition Notation I Background Development 1 Semigroups and Their Ideals 1.1 Semigroups 1.1.1 Partial 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 1.8 Notes 2 Right Topological (and Semitopological and Topological) Semigroups 2.1 Topological Hierarchy 2.2 Compact Right Topological Semigroups 2.3 Closures and Products of Ideals 2.4 Semitopological and Topological Semigroups 2.5 Ellis’ Theorem 2.6 Notes 3 ßD -Ultrafilters and The Stone-Čech Compactification of a Discrete Space 3.1 Ultrafilters 3.2 The Topological Space ßD 3.3 Stone–Čech Compactification 3.4 More Topology of ßD 3.5 Uniform Limits via Ultrafilters 3.6 The Cardinality of ßD 3.7 Notes 3.8 Closing Remarks 4 ßS – The Stone-Čech Compactification of a Discrete Semigroup 4.1 Extending the Operation to ßS 4.2 Commutativity in ßS 4.3 S * 4.4 K(ßS) and its Closure 4.5 Notions of Size 4.6 Notes 5 ßS and Ramsey Theory – Some Easy Applications 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 5.6 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 – Copies of ℍ 6.4 Weakly Left Cancellative Semigroups 6.5 Semiprincipal Left Ideals and the Center of p(ßS)p 6.6 Principal Ideals in ßℤ 6.7 Ideals and Density 6.8 Notes 7 Groups in ßS 7.1 Zelenyuk’s Theorem 7.2 Semigroups Isomorphic to ℍ 7.3 Free Semigroups and Free Groups in ßS 7.4 Discrete copies of ℤ 7.5 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 Determined by Right Cancelable Elements in Countable Groups 8.6 Notes 9 Idempotents 9.1 Right Maximal Idempotents 9.2 Topologies Defined by Idempotents 9.3 Chains of Idempotents 9.4 Identities in ßS 9.5 Rectangular Semigroups in ßℕ 9.6 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 Defined on Principal Left and Right Ideals 10.5 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 11.5 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 of Strongly Summable Ultrafilters 12.5 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 The Left and Right Continuous Extensions of One Operation 13.5 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 A Combinatorial Characterization of Central Sets 14.6 Geoarithmetic Progressions 14.7 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 – Rado’s Theorem 15.4 Image Partition Regularity over ℕ 15.5 Matrices with Entries from Fields 15.6 Infinite Image Partition Regular Matrices 15.7 Notes 16 IP, IP*, Central, and Central* Sets 16.1 IP, IP*, Central, and Central* 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 16.5 Notes 17 Sums and Products 17.1 Ultrafilters with Rich Additive and Multiplicative Structure 17.2 Pairwise Sums and Products 17.3 Sums of Products 17.4 Linear Combinations of Sums - Infinite Partition Regular Matrices 17.5 Sums and Products in (0,1) – Measurable and Baire Partitions 17.6 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 18.5 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 19.5 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 20.4 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 Semigroup Compactifications as Spaces of Filters 21.5 Uniform Compactifications 21.6 Notes Bibliography Index