Partially Ordered Abelian Groups With Interpolation
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Partially Ordered Abelian Groups With Interpolation
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Author : Kenneth R. Goodearl
language : en
Publisher: American Mathematical Soc.
Release Date : 2010
Partially Ordered Abelian Groups With Interpolation written by Kenneth R. Goodearl 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.
A branch of ordered algebraic structures has grown, motivated by $K$-theoretic applications and mainly concerned with partially ordered abelian groups satisfying the Riesz interpolation property. This monograph is the first source in which the algebraic and analytic aspects of these interpolation groups have been integrated into a coherent framework for general reference. The author provides a solid foundation in the structure theory of interpolation groups and dimension groups (directed unperforated interpolation groups), with applications to ordered $K$-theory particularly in mind. Although interpolation groups are defined as purely algebraic structures, their development has been strongly influenced by functional analysis. This cross-cultural development has left interpolation groups somewhat estranged from both the algebraists, who may feel intimidated by compact convex sets, and the functional analysts, who may feel handicapped by the lack of scalars. This book, requiring only standard first-year graduate courses in algebra and functional analysis, aims to make the subject accessible to readers from both disciplines. High points of the development include the following: characterization of dimension groups as direct limits of finite products of copies of the integers; the double-dual representation of an interpolation group with order-unit via affine continuous real-valued functions on its state space; the structure of dimension groups complete with respect to the order-unit norm, as well as monotone sigma-complete dimension groups and dimension groups with countably infinite interpolation; and an introduction to the problem of classifying extensions of one dimension group by another. The book also includes a development of portions of the theory of compact convex sets and Choquet simplices, and an expository discussion of various applications of interpolation group theory to rings and $C^*$-algebras via ordered $K_0$. A discussion of some open problems in interpolation groups and dimension groups concludes the book. Of interest, of course, to researchers in ordered algebraic structures, the book will also be a valuable source for researchers seeking a background in interpolation groups and dimension groups for applications to such subjects as rings, operator algebras, topological Markov chains, positive polynomials, compact group actions, or other areas where ordered Grothendieck groups might be useful. This is a reprint of the 1986 original. (SURV/20.S)
Partially Ordered Groups
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Author : A M W Glass
language : en
Publisher: World Scientific
Release Date : 1999-07-22
Partially Ordered Groups written by A M W Glass and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-07-22 with Mathematics categories.
Recently the theory of partially ordered groups has been used by analysts, algebraists, topologists and model theorists. This book presents the most important results and topics in the theory with proofs that rely on (and interplay with) other areas of mathematics. It concludes with a list of some unsolved problems for the reader to tackle. In stressing both the special techniques of the discipline and the overlap with other areas of pure mathematics, the book should be of interest to a wide audience in diverse areas of mathematics. Contents:Definitions and ExamplesBasic PropertiesValues, Primes and PolarsAbelian and Normal-Valued Lattice-Ordered GroupsArchimedean Function GroupsSoluble Right Partially Ordered Groups and GeneralisationsPermutationsApplicationsCompletionsVarieties of Lattice-Ordered GroupsUnsolved Problems Readership: Pure mathematicians. Keywords:Partially Ordered Group;Lattice Ordered Group;Abelian Lattice Ordered Group;Completion;VarietyReviews: “The author's style of writing is very lucid, and the material presented is self-contained. It is an excellent reference text for a graduate course in this area, as well as a source of material for individual reading.” Bulletin of London Mathematical Society “This monograph is clearly written, well organized … can be warmly recommended to students and research workers dealing with the theory of partially ordered groups.” Mathematics Abstracts “Glass's book will get the reader to the forefront of research in the field and would be a suitable text for students in modern algebra, group theory, or ordered structures. It will surely find its place in all mathematical libraries and on the desks of the professional algebraists and 'ordered-groupers'.” Mathematical Reviews
Abelian Groups And Representations Of Finite Partially Ordered Sets
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Author : David Arnold
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-11-14
Abelian Groups And Representations Of Finite Partially Ordered Sets written by David Arnold 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 2012-11-14 with Mathematics categories.
The theme of this book is an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible. David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals.
The Theory Of Lattice Ordered Groups
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Author : V.M. Kopytov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
The Theory Of Lattice Ordered Groups written by V.M. Kopytov 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 2013-03-09 with Mathematics categories.
A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam ple, partially ordered groups with interpolation property were intro duced in F. Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P.
Affine Representations Of Grothendieck Groups And Applications To Rickart C Algebras And Aleph 0 Continuous Regular Rings
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Author : K. R. Goodearl
language : en
Publisher: American Mathematical Soc.
Release Date : 1980
Affine Representations Of Grothendieck Groups And Applications To Rickart C Algebras And Aleph 0 Continuous Regular Rings written by K. R. Goodearl 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 1980 with Mathematics categories.
Algebra And Its Applications
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Author : H. L. Manocha
language : en
Publisher: CRC Press
Release Date : 1984-05-31
Algebra And Its Applications written by H. L. Manocha and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-05-31 with Mathematics categories.
This volume unites more than fifty international mathematicians, spotlighting research that demonstrates the importance of algebra in science and engineering. Areas in algebra such as invariant theory, group representations, commutative algebra, and algebraic geometry are important factors in such subjects as quantum physics, computing, and data communications. The International Symposium on Algebra and Its Applications was organized by the Department of Mathematics of the Indian Institute of Technology, and held in New Delhi, India, December 21-25, 1981. This volume contains papers presented, and the editors wish to express their appreciation to all the authors for their submissions, and symposium participants for their enthusiasm.
Refinement Monoids Equidecomposability Types And Boolean Inverse Semigroups
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Author : Friedrich Wehrung
language : en
Publisher: Springer
Release Date : 2017-09-09
Refinement Monoids Equidecomposability Types And Boolean Inverse Semigroups written by Friedrich Wehrung and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-09 with Mathematics categories.
Adopting a new universal algebraic approach, this book explores and consolidates the link between Tarski's classical theory of equidecomposability types monoids, abstract measure theory (in the spirit of Hans Dobbertin's work on monoid-valued measures on Boolean algebras) and the nonstable K-theory of rings. This is done via the study of a monoid invariant, defined on Boolean inverse semigroups, called the type monoid. The new techniques contrast with the currently available topological approaches. Many positive results, but also many counterexamples, are provided.
The Complete Dimension Theory Of Partially Ordered Systems With Equivalence And Orthogonality
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Author : K. R. Goodearl
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
The Complete Dimension Theory Of Partially Ordered Systems With Equivalence And Orthogonality written by K. R. Goodearl 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 2005 with Mathematics categories.
Introduction Partial commutative monoids Continuous dimension scales Espaliers Classes of espaliers Bibliography Index
Current Research In Operational Quantum Logic
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Author : Bob Coecke
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Current Research In Operational Quantum Logic written by Bob Coecke 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 2013-06-29 with Science categories.
The present volume has its origins in a pair of informal workshops held at the Free University of Brussels, in June of 1998 and May of 1999, named "Current Research 1 in Operational Quantum Logic". These brought together mathematicians and physicists working in operational quantum logic and related areas, as well as a number of interested philosophers of science, for a rare opportunity to discuss recent developments in this field. After some discussion, it was decided that, rather than producing a volume of conference proceedings, we would try to organize the conferees to produce a set of comprehensive survey papers, which would not only report on recent developments in quantum logic, but also provide a tutorial overview of the subject suitable for an interested non-specialist audience. The resulting volume provides an overview of the concepts and methods used in current research in quantum logic, viewed both as a branch of mathemati cal physics and as an area of pure mathematics. The first half of the book is concerned with the algebraic side of the subject, and in particular the theory of orthomodular lattices and posets, effect algebras, etc. In the second half of the book, special attention is given to categorical methods and to connections with theoretical computer science. At the 1999 workshop, we were fortunate to hear three excellent lectures by David J. Foulis, represented here by two contributions. Dave's work, spanning 40 years, has helped to define, and continues to reshape, the field of quantum logic.
Commutative Semigroups
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Author : P.A. Grillet
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Commutative Semigroups written by P.A. Grillet 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 2013-06-29 with Mathematics categories.
This is the first book about commutative semigroups in general. Emphasis is on structure but the other parts of the theory are at least surveyed and a full set of about 850 references is included. The book is intended for mathematicians who do research on semigroups or who encounter commutative semigroups in their research.