Ultrafilters Throughout Mathematics
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Ultrafilters Throughout Mathematics
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Author : Isaac Goldbring
language : en
Publisher: American Mathematical Society
Release Date : 2022-06-13
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 2022-06-13 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.
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 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.
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
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Author : 黃毅英
language : en
Publisher:
Release Date : 2017-01-27
Ultrafilters And Topologies written by 黃毅英 and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-27 with categories.
Crowded Rational Ultrafilters
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Author : Eva Coplakova
language : en
Publisher:
Release Date : 1996
Crowded Rational Ultrafilters written by Eva Coplakova and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with categories.
Models And Ultraproducts
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Author : John Lane Bell
language : en
Publisher: Courier Corporation
Release Date : 2006-01-01
Models And Ultraproducts written by John Lane Bell and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-01 with Mathematics categories.
In this text for first-year graduate students, the authors provide an elementary exposition of some of the basic concepts of model theory--focusing particularly on the ultraproduct construction and the areas in which it is most useful. The book, which assumes only that its readers are acquainted with the rudiments of set theory, starts by developing the notions of Boolean algebra, propositional calculus, and predicate calculus. Model theory proper begins in the fourth chapter, followed by an introduction to ultraproduct construction, which includes a detailed look at its theoretic properties. An overview of elementary equivalence provides algebraic descriptions of the elementary classes. Discussions of completeness follow, along with surveys of the work of Jónsson and of Morley and Vaught on homogeneous universal models, and the results of Keisler in connection with the notion of a saturated structure. Additional topics include classical results of Gödel and Skolem, and extensions of classical first-order logic in terms of generalized quantifiers and infinitary languages. Numerous exercises appear throughout the text.
The Use Of Ultraproducts In Commutative Algebra
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Author : Hans Schoutens
language : en
Publisher: Springer
Release Date : 2010-07-16
The Use Of Ultraproducts In Commutative Algebra written by Hans Schoutens and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-16 with Mathematics categories.
In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
Ultrafilters
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Author : D. G. Glavasic
language : en
Publisher:
Release Date : 1990
Ultrafilters written by D. G. Glavasic and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Set theory categories.
