Categorical Structure Of Closure Operators

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Categorical Structure of Closure Operators

D. Dikranjan,Walter Tholen
Categorical Structure of Closure Operators Book PDF

Author : D. Dikranjan,Walter Tholen
Publisher : Springer Science & Business Media
Page : 373 pages
File Size : 54,8 Mb
Release : 2013-04-09
Category : Mathematics
ISBN : 9789401584005

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Categorical Structure of Closure Operators by D. Dikranjan,Walter Tholen Pdf

Our motivation for gathering the material for this book over aperiod of seven years has been to unify and simplify ideas wh ich appeared in a sizable number of re search articles during the past two decades. More specifically, it has been our aim to provide the categorical foundations for extensive work that was published on the epimorphism- and cowellpoweredness problem, predominantly for categories of topological spaces. In doing so we found the categorical not ion of closure operators interesting enough to be studied for its own sake, as it unifies and describes other significant mathematical notions and since it leads to a never-ending stream of ex amples and applications in all areas of mathematics. These are somewhat arbitrarily restricted to topology, algebra and (a small part of) discrete mathematics in this book, although other areas, such as functional analysis, would provide an equally rich and interesting supply of examples. We also had to restrict the themes in our theoretical exposition. In spite of the fact that closure operators generalize the uni versal closure operations of abelian category theory and of topos- and sheaf theory, we chose to mention these aspects only en passant, in favour of the presentation of new results more closely related to our original intentions. We also needed to refrain from studying topological concepts, such as compactness, in the setting of an arbitrary closure-equipped category, although this topic appears prominently in the published literature involving closure operators.

Categorical Closure Operators

Gabriele Castellini
Categorical Closure Operators Book PDF

Author : Gabriele Castellini
Publisher : Springer Science & Business Media
Page : 300 pages
File Size : 52,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817682347

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Categorical Closure Operators by Gabriele Castellini Pdf

This book presents the general theory of categorical closure operators to gether with a number of examples, mostly drawn from topology and alge bra, which illustrate the general concepts in several concrete situations. It is aimed mainly at researchers and graduate students in the area of cate gorical topology, and to those interested in categorical methods applied to the most common concrete categories. Categorical Closure Operators is self-contained and can be considered as a graduate level textbook for topics courses in algebra, topology or category theory. The reader is expected to have some basic knowledge of algebra, topology and category theory, however, all categorical concepts that are recurrent are included in Chapter 2. Moreover, Chapter 1 contains all the needed results about Galois connections, and Chapter 3 presents the the ory of factorization structures for sinks. These factorizations not only are essential for the theory developed in this book, but details about them can not be found anywhere else, since all the results about these factorizations are usually treated as the duals of the theory of factorization structures for sources. Here, those hard-to-find details are provided. Throughout the book I have kept the number of assumptions to a min imum, even though this implies that different chapters may use different hypotheses. Normally, the hypotheses in use are specified at the beginning of each chapter and they also apply to the exercise set of that chapter.

Categorical Structure of Closure Operators

Dikran Dikranjan,Walter Tholen
Categorical Structure of Closure Operators Book PDF

Author : Dikran Dikranjan,Walter Tholen
Publisher : Springer
Page : 358 pages
File Size : 40,7 Mb
Release : 2013-01-08
Category : Mathematics
ISBN : 940158401X

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Categorical Structure of Closure Operators by Dikran Dikranjan,Walter Tholen Pdf

Our motivation for gathering the material for this book over aperiod of seven years has been to unify and simplify ideas wh ich appeared in a sizable number of re search articles during the past two decades. More specifically, it has been our aim to provide the categorical foundations for extensive work that was published on the epimorphism- and cowellpoweredness problem, predominantly for categories of topological spaces. In doing so we found the categorical not ion of closure operators interesting enough to be studied for its own sake, as it unifies and describes other significant mathematical notions and since it leads to a never-ending stream of ex amples and applications in all areas of mathematics. These are somewhat arbitrarily restricted to topology, algebra and (a small part of) discrete mathematics in this book, although other areas, such as functional analysis, would provide an equally rich and interesting supply of examples. We also had to restrict the themes in our theoretical exposition. In spite of the fact that closure operators generalize the uni versal closure operations of abelian category theory and of topos- and sheaf theory, we chose to mention these aspects only en passant, in favour of the presentation of new results more closely related to our original intentions. We also needed to refrain from studying topological concepts, such as compactness, in the setting of an arbitrary closure-equipped category, although this topic appears prominently in the published literature involving closure operators.

Categorical Structures and Their Applications

Werner G„hler,Gerhard Preuss
Categorical Structures and Their Applications Book PDF

Author : Werner G„hler,Gerhard Preuss
Publisher : World Scientific
Page : 356 pages
File Size : 44,8 Mb
Release : 2004
Category : Mathematics
ISBN : 9789812560537

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Categorical Structures and Their Applications by Werner G„hler,Gerhard Preuss Pdf

The book collects original research papers on applied categorical structures, most of which have been presented at the North-West European Category Seminar 2003 in Berlin. The spectrum of these mathematical results reflects the varied interests of Horst Herrlich ? one of the leading category theorists of the world ? to whom this volume is dedicated in view of his 65th birthday. The book contains applications of categorical methods in various branches of mathematics such as algebra, analysis, logic and topology, as well as fuzzy structures and computer science. At the end of the book the reader will find a complete list of Horst Herrlich's publications.The proceedings have been selected for coverage in: ? Index to Scientific & Technical Proceedings? (ISTP? / ISI Proceedings)? Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)? CC Proceedings ? Engineering & Physical Sciences

Categorical Structures and Their Applications

W Gähler,G Preuss
Categorical Structures and Their Applications Book PDF

Author : W Gähler,G Preuss
Publisher : World Scientific
Page : 356 pages
File Size : 48,6 Mb
Release : 2004-09-28
Category : Mathematics
ISBN : 9789814481700

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Categorical Structures and Their Applications by W Gähler,G Preuss Pdf

The book collects original research papers on applied categorical structures, most of which have been presented at the North-West European Category Seminar 2003 in Berlin. The spectrum of these mathematical results reflects the varied interests of Horst Herrlich — one of the leading category theorists of the world — to whom this volume is dedicated in view of his 65th birthday. The book contains applications of categorical methods in various branches of mathematics such as algebra, analysis, logic and topology, as well as fuzzy structures and computer science. At the end of the book the reader will find a complete list of Horst Herrlich's publications. The proceedings have been selected for coverage in: • Index to Scientific & Technical Proceedings® (ISTP® / ISI Proceedings) • Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings) • CC Proceedings — Engineering & Physical Sciences Contents:On Boole's Booleanness (B Banaschewski)Homeomorphically Closed Nearness Spaces (H L Bentley & J W Carlson)The Tensor Product of Orthomodular Posets (R Börger)Saturated Collections of Metrics (E Colebunders et al.)Topological Structures in Logics (W Gähler)The Structure of Affine Algebraic Sets (E Giuli)A Characterization of Co-Retracts of Functional Structures (R Kaschek)Introducing Lagois Correspondences (A Melton)On Coalgebras which are Algebras (H-E Porst & C Dzierzon)Fibrewise Sobriety (G Richter & A Vauth)and other papers Readership: Graduate students, researchers, academics and lecturers in mathematics and computer science. Keywords:Category Theory;Algebra;Logic;Topology;Fuzzy Structures;Computer Science

Categorical Perspectives

Jürgen Koslowski,Austin Melton
Categorical Perspectives Book PDF

Author : Jürgen Koslowski,Austin Melton
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 46,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461213703

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Categorical Perspectives by Jürgen Koslowski,Austin Melton Pdf

"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas. Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preuá * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina

Jordan, Real and Lie Structures in Operator Algebras

Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov
Jordan Real and Lie Structures in Operator Algebras Book PDF

Author : Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov
Publisher : Springer Science & Business Media
Page : 239 pages
File Size : 44,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401586054

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Jordan, Real and Lie Structures in Operator Algebras by Sh. Ayupov,Abdugafur Rakhimov,Shukhrat Usmanov Pdf

The theory of operator algebras acting on a Hilbert space was initiated in thirties by papers of Murray and von Neumann. In these papers they have studied the structure of algebras which later were called von Neu mann algebras or W* -algebras. They are weakly closed complex *-algebras of operators on a Hilbert space. At present the theory of von Neumann algebras is a deeply developed theory with various applications. In the framework of von Neumann algebras theory the study of fac tors (i.e. W* -algebras with trivial centres) is very important, since they are comparatively simple and investigation of general W* -algebras can be reduced to the case of factors. Therefore the theory of factors is one of the main tools in the structure theory of von Neumann algebras. In the middle of sixtieth Topping [To 1] and Stormer [S 2] have ini tiated the study of Jordan (non associative and real) analogues of von Neumann algebras - so called JW-algebras, i.e. real linear spaces of self adjoint opera.tors on a complex Hilbert space, which contain the identity operator 1. closed with respect to the Jordan (i.e. symmetrised) product INTRODUCTION 2 x 0 y = ~(Xy + yx) and closed in the weak operator topology. The structure of these algebras has happened to be close to the struc ture of von Neumann algebras and it was possible to apply ideas and meth ods similar to von Neumann algebras theory in the study of JW-algebras.

Categorical Topology

Eraldo Giuli
Categorical Topology Book PDF

Author : Eraldo Giuli
Publisher : Springer Science & Business Media
Page : 281 pages
File Size : 40,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400902633

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Categorical Topology by Eraldo Giuli Pdf

Proceedings of the L'Aquila Conference 1994

Categorical Foundations

Maria Cristina Pedicchio,Walter Tholen
Categorical Foundations Book PDF

Author : Maria Cristina Pedicchio,Walter Tholen
Publisher : Cambridge University Press
Page : 452 pages
File Size : 42,7 Mb
Release : 2004
Category : Mathematics
ISBN : 0521834147

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Categorical Foundations by Maria Cristina Pedicchio,Walter Tholen Pdf

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Algebraic Structures and Operators Calculus

P. Feinsilver,René Schott
Algebraic Structures and Operators Calculus Book PDF

Author : P. Feinsilver,René Schott
Publisher : Springer Science & Business Media
Page : 246 pages
File Size : 52,5 Mb
Release : 1993
Category : Computers
ISBN : 0792338340

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Algebraic Structures and Operators Calculus by P. Feinsilver,René Schott Pdf

Introduction I. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 III. Lie algebras: some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter 1 Operator calculus and Appell systems I. Boson calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 II. Holomorphic canonical calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 III. Canonical Appell systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 2 Representations of Lie groups I. Coordinates on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 II. Dual representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 III. Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 IV. Induced representations and homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 General Appell systems Chapter 3 I. Convolution and stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 II. Stochastic processes on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 III. Appell systems on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 4 Canonical systems in several variables I. Homogeneous spaces and Cartan decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 II. Induced representation and coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 III. Orthogonal polynomials in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter 5 Algebras with discrete spectrum I. Calculus on groups: review of the theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 II. Finite-difference algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 III. q-HW algebra and basic hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 IV. su2 and Krawtchouk polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 V. e2 and Lommel polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 6 Nilpotent and solvable algebras I. Heisenberg algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 II. Type-H Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Vll III. Upper-triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 IV. Affine and Euclidean algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Chapter 7 Hermitian symmetric spaces I. Basic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 II. Space of rectangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 III. Space of skew-symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 IV. Space of symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 8 Properties of matrix elements I. Addition formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 II. Recurrences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 III. Quotient representations and summation formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 9 Symbolic computations I. Computing the pi-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 II. Adjoint group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 III. Recursive computation of matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Papers in Honour of Bernhard Banaschewski

Guillaume Brümmer,Christopher Gilmour
Papers in Honour of Bernhard Banaschewski Book PDF

Author : Guillaume Brümmer,Christopher Gilmour
Publisher : Springer Science & Business Media
Page : 496 pages
File Size : 48,6 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9789401725293

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Papers in Honour of Bernhard Banaschewski by Guillaume Brümmer,Christopher Gilmour Pdf

Proceedings of the BB Fest 96, a conference held at the University of Cape Town, 15-20 July 1996, on Category Theory and its Applications to Topology, Order and Algebra

Monoidal Topology

Dirk Hofmann,Gavin J. Seal,Walter Tholen
Monoidal Topology Book PDF

Author : Dirk Hofmann,Gavin J. Seal,Walter Tholen
Publisher : Cambridge University Press
Page : 523 pages
File Size : 46,7 Mb
Release : 2014-07-31
Category : Mathematics
ISBN : 9781107063945

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Monoidal Topology by Dirk Hofmann,Gavin J. Seal,Walter Tholen Pdf

Based on lax-algebraic and categorical methods, Monoidal Topology provides a unified theory for metric and topological structures with far-reaching applications.

Beyond Topology

FrŽdŽric Mynard,Elliott Pearl
Beyond Topology Book PDF

Author : FrŽdŽric Mynard,Elliott Pearl
Publisher : American Mathematical Soc.
Page : 395 pages
File Size : 45,8 Mb
Release : 2009-05-15
Category : Mathematics
ISBN : 9780821842799

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Beyond Topology by FrŽdŽric Mynard,Elliott Pearl Pdf

The purpose of this collection is to guide the non-specialist through the basic theory of various generalizations of topology, starting with clear motivations for their introduction. Structures considered include closure spaces, convergence spaces, proximity spaces, quasi-uniform spaces, merotopic spaces, nearness and filter spaces, semi-uniform convergence spaces, and approach spaces. Each chapter is self-contained and accessible to the graduate student, and focuses on motivations to introduce the generalization of topologies considered, presenting examples where desirable properties are not present in the realm of topologies and the problem is remedied in the more general context. Then, enough material will be covered to prepare the reader for more advanced papers on the topic. While category theory is not the focus of the book, it is a convenient language to study these structures and, while kept as a tool rather than an object of study, will be used throughout the book. For this reason, the book contains an introductory chapter on categorical topology.

Rings, Modules, and Closure Operations

Jesse Elliott
Rings Modules and Closure Operations Book PDF

Author : Jesse Elliott
Publisher : Springer Nature
Page : 490 pages
File Size : 51,8 Mb
Release : 2019-11-30
Category : Mathematics
ISBN : 9783030244019

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Rings, Modules, and Closure Operations by Jesse Elliott Pdf

This book presents a systematic exposition of the various applications of closure operations in commutative and noncommutative algebra. In addition to further advancing multiplicative ideal theory, the book opens doors to the various uses of closure operations in the study of rings and modules, with emphasis on commutative rings and ideals. Several examples, counterexamples, and exercises further enrich the discussion and lend additional flexibility to the way in which the book is used, i.e., monograph or textbook for advanced topics courses.

Progress in Commutative Algebra 2

Christopher Francisco,Lee C. Klingler,Sean M. Sather-Wagstaff,Janet C. Vassilev
Progress in Commutative Algebra 2 Book PDF

Author : Christopher Francisco,Lee C. Klingler,Sean M. Sather-Wagstaff,Janet C. Vassilev
Publisher : Walter de Gruyter
Page : 329 pages
File Size : 51,6 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9783110278606

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Progress in Commutative Algebra 2 by Christopher Francisco,Lee C. Klingler,Sean M. Sather-Wagstaff,Janet C. Vassilev Pdf

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. However, in the study of non-noetherian rings it is much easier to find a ring having a finite number of prime ideals. The editors have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. The editors have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization.