Ultrafilters
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The Theory Of Ultrafilters
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Author : W.W. Comfort
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
The Theory Of Ultrafilters written by W.W. Comfort and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
An ultrafilter is a truth-value assignment to the family of subsets of a set, and a method of convergence to infinity. From the first (logical) property arises its connection with two-valued logic and model theory; from the second (convergence) property arises its connection with topology and set theory. Both these descriptions of an ultrafilter are connected with compactness. The model-theoretic property finds its expression in the construction of the ultraproduct and the compactness type of theorem of Los (implying the compactness theorem of first-order logic); and the convergence property leads to the process of completion by the adjunction of an ideal element for every ultrafilter-i. e. , to the Stone-Cech com pactification process (implying the Tychonoff theorem on the compact ness of products). Since these are two ways of describing the same mathematical object, it is reasonable to expect that a study of ultrafilters from these points of view will yield results and methods which can be fruitfully crossbred. This unifying aspect is indeed what we have attempted to emphasize in the present work.
Ultrafilters And Topologies On Groups
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Author : Yevhen Zelenyuk
language : en
Publisher: Walter de Gruyter
Release Date : 2011-03-29
Ultrafilters And Topologies On Groups written by Yevhen Zelenyuk and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-03-29 with Mathematics categories.
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.
Ultrafilters Throughout Mathematics
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Author : Isaac Goldbring
language : en
Publisher: American Mathematical Society
Release Date : 2022-06-28
Ultrafilters Throughout Mathematics written by Isaac Goldbring and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-06-28 with Mathematics categories.
Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty.
Ultrafilters Across Mathematics
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Author : Vitaly Bergelson
language : en
Publisher: American Mathematical Soc.
Release Date : 2010
Ultrafilters Across Mathematics written by Vitaly Bergelson and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
Presents the state-of-the-art of applications in the whole spectrum of mathematics which are grounded on the use of ultrafilters and ultraproducts. It contains two general surveys on ultrafilters in set theory and on the ultraproduct construction, as well as papers that cover additive and combinatorial number theory, nonstandard methods and stochastic differential equations, measure theory, dynamics, Ramsey theory, algebra in the space of ultrafilters, and large cardinals.
Everything You Wanted To Know About Ultrafilters But Were Afraid To Ask
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Author : Jussi Ketonen
language : en
Publisher:
Release Date : 1971
Everything You Wanted To Know About Ultrafilters But Were Afraid To Ask written by Jussi Ketonen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971 with Combinatorial set theory categories.
Countably Indexed Ultrafilters
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Author : David Douglas Booth
language : en
Publisher:
Release Date : 1969
Countably Indexed Ultrafilters written by David Douglas Booth and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with Logic, Symbolic and mathematical categories.
Some Topological Results On Ultrafilters
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Author : Joni Baker
language : en
Publisher:
Release Date : 2001
Some Topological Results On Ultrafilters written by Joni Baker and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with categories.
Nonstandard Analysis And Special Ultrafilters
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Author : Michael Abraham Benedikt
language : en
Publisher:
Release Date : 1993
Nonstandard Analysis And Special Ultrafilters written by Michael Abraham Benedikt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993 with categories.
Filters And Ultrafilters Over Definable Subsets Of Admissible Ordinals
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Author : J. C. M. Baeten
language : en
Publisher:
Release Date : 1986
Filters And Ultrafilters Over Definable Subsets Of Admissible Ordinals written by J. C. M. Baeten and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1986 with Admissible sets categories.
Ultrafilters And Topologies On Groups
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Author : Yevhen G. Zelenyuk
language : en
Publisher: Walter de Gruyter
Release Date : 2011
Ultrafilters And Topologies On Groups written by Yevhen G. Zelenyuk and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
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 22G 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.
