Ultrafilters and Topologies on Groups (De Gruyter Expositions in Mathematics, 50)

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification __βG__ of a discrete group __G__ is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if __G__ is a countable torsion free group, then __βG__ contains no nontrivial finite groups. Also the ideal structure of __βG__ is investigated. In particular, one shows that for every infinite Abelian group __G__, __βG__ contains 2^2^|G| minimal right ideals. In the third part, using the semigroup __βG__, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely __ω__-resolvable, and consequently, can be partitioned into __ω__ subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas. * Presents the relationship between ultrafilters and topologies on groups

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters.

The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous.

In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 22|G|minimal right ideals.

In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition.

The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.

This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G , βG contains 2 2 | G | minimal right ideals. In the third part, using the semigroup βG , almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω -resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas. Presents the relationship between ultrafilters and topologies on groups Preface Contents 1 Topological Groups 1.1 The Notion of a Topological Group 1.2 The Neighborhood Filter of the Identity 1.3 The Topology T (F) 1.4 Topologizing a Group 1.5 Metrizable Refinements 1.6 Topologizability of a Countably Infinite Ring 2 Ultrafilters 2.1 The Notion of an Ultrafilter 2.2 The Space βD 2.3 Martin’s Axiom 2.4 Ramsey Ultrafilters and P -points 2.5 Measurable Cardinals 3 Topological Spaces with Extremal Properties 3.1 Filters and Ultrafilters on Topological Spaces 3.2 Spaces with Extremal Properties 3.3 Irresolvability 4 Left Invariant Topologies and Strongly Discrete Filters 4.1 Left Topological Semigroups 4.2 The Topology T[F] 4.3 Strongly Discrete Filters 4.4 Invariant Topologies 5 Topological Groups with Extremal Properties 5.1 Extremally Disconnected Topological Groups 5.2 Maximal Topological Groups 5.3 Nodec Topological Groups 5.4 P -point Theorems 6 The Semigroup βS 6.1 Extending the Operation to βS 6.2 Compact Right Topological Semigroups 6.3 Hindman’s Theorem 6.4 Ultrafilters from K(βS) 7 Ultrafilter Semigroups 7.1 The Semigroup Ult(T) 7.2 Regularity 7.3 Homomorphisms 8 Finite Groups in βG 8.1 Local Left Groups and Local Homomorphisms 8.2 Triviality of Finite Groups in βZ 8.3 Local Automorphisms of Finite Order 8.4 Finite Groups in G* 9 Ideal Structure of βG 9.1 Left Ideals 9.2 Right Ideals 9.3 The Structure Group of K(βG) 9.4 K (βG) is not Closed 10 Almost Maximal Topological Groups 10.1 Construction 10.2 Properties 10.3 Semilattice Decompositions and Burnside Semigroups 10.4 Projectives 10.5 Topological Invariantness of Ult(T) 11 Almost Maximal Spaces 11.1 Right Maximal Idempotents in HK 11.2 Projectivity of Ult(T) 11.3 The Semigroup C(p) 11.4 Local Monomorphisms 12 Resolvability 12.1 Regular Homeomorphisms of Finite Order 12.2 Resolvability of Topological Groups 12.3 Absolute Resolvability 13 Open Problems Bibliography Index This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. Topics covered include: topological and left topological groups, ultrafilter semigroups, local homomorphisms and automorphisms, subgroups and ideal structure of ßG, almost maximal spaces and projectives of finite semigroups, resolvability of groups. This is a self-contained book aimed at graduate students and researchers working in topological algebra and adjacent areas. From the contents: Topological Groups Ultrafilters Topological Spaces with Extremal Properties Left Invariant Topologies and Strongly Discrete Filters Topological Groups with Extremal Properties The Semigroup ßS Ultrafilter Semigroups Finite Groups in ßG Ideal Structure of ßS Almost Maximal Topological Groups and Spaces Resolvability Open Problems From the contents: Topological Groups Ultrafilters Topological Spaces with Extremal Properties Left Invariant Topologies and Strongly Discrete Filters Topological Groups with Extremal Properties The Semigroup ssS Ultrafilter Semigroups Finite Groups in ssG Ideal Structure of ssS Almost Maximal Topological Groups and Spaces Resolvability Open Problems