The Theory Of Lattice Ordered Groups

Author : M.E Anderson,T.H. Feil
Publisher : Springer Science & Business Media
Page : 197 pages
File Size : 46,7 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9789400928718

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Lattice-Ordered Groups by M.E Anderson,T.H. Feil Pdf

The study of groups equipped with a compatible lattice order ("lattice-ordered groups" or "I!-groups") has arisen in a number of different contexts. Examples of this include the study of ideals and divisibility, dating back to the work of Dedekind and continued by Krull; the pioneering work of Hahn on totally ordered abelian groups; and the work of Kantorovich and other analysts on partially ordered function spaces. After the Second World War, the theory of lattice-ordered groups became a subject of study in its own right, following the publication of fundamental papers by Birkhoff, Nakano and Lorenzen. The theory blossomed under the leadership of Paul Conrad, whose important papers in the 1960s provided the tools for describing the structure for many classes of I!-groups in terms of their convex I!-subgroups. A particularly significant success of this approach was the generalization of Hahn's embedding theorem to the case of abelian lattice-ordered groups, work done with his students John Harvey and Charles Holland. The results of this period are summarized in Conrad's "blue notes" [C].

Author : Michael Darnel
Publisher : CRC Press
Page : 554 pages
File Size : 41,7 Mb
Release : 2021-12-16
Category : Mathematics
ISBN : 9781000105179

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Theory of Lattice-Ordered Groups by Michael Darnel Pdf

Provides a thorough discussion of the orderability of a group. The book details the major developments in the theory of lattice-ordered groups, delineating standard approaches to structural and permutation representations. A radically new presentation of the theory of varieties of lattice-ordered groups is offered.;This work is intended for pure and applied mathematicians and algebraists interested in topics such as group, order, number and lattice theory, universal algebra, and representation theory; and upper-level undergraduate and graduate students in these disciplines.;College or university bookstores may order five or more copies at a special student price which is available from Marcel Dekker Inc, upon request.

Author : V.M. Kopytov,N.Ya. Medvedev
Publisher : Springer
Page : 400 pages
File Size : 52,6 Mb
Release : 2013-01-07
Category : Mathematics
ISBN : 9401583056

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The Theory of Lattice-Ordered Groups by V.M. Kopytov,N.Ya. Medvedev Pdf

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.

Author : V.M. Kopytov,N.Ya. Medvedev
Publisher : Springer Science & Business Media
Page : 426 pages
File Size : 44,8 Mb
Release : 1994-10-31
Category : Mathematics
ISBN : 0792331699

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The Theory of Lattice-Ordered Groups by V.M. Kopytov,N.Ya. Medvedev Pdf

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.

Author : A M W Glass
Publisher : World Scientific
Page : 324 pages
File Size : 51,8 Mb
Release : 1999-07-22
Category : Mathematics
ISBN : 9789814496094

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Partially Ordered Groups by A M W Glass Pdf

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

Author : A.M. Glass,W.C. Holland
Publisher : Springer Science & Business Media
Page : 398 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400922839

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Lattice-Ordered Groups by A.M. Glass,W.C. Holland Pdf

A lattice-ordered group is a mathematical structure combining a (partial) order (lattice) structure and a group structure (on a set) in a compatible way. Thus it is a composite structure, or, a set carrying two or more simple structures in a compatible way. The field of lattice-ordered groups turn up on a wide range of mathematical fields ranging from functional analysis to universal algebra. These papers address various aspects of the field, with wide applicability for interested researchers.

Author : V.M. Kopytov,N.Ya. Medvedev
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 49,8 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401583046

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The Theory of Lattice-Ordered Groups by V.M. Kopytov,N.Ya. Medvedev Pdf

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.

Author : Michael Darnel
Publisher : CRC Press
Page : 568 pages
File Size : 49,9 Mb
Release : 2021-12-17
Category : Mathematics
ISBN : 9781000148381

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Theory of Lattice-Ordered Groups by Michael Darnel Pdf

Provides a thorough discussion of the orderability of a group. The book details the major developments in the theory of lattice-ordered groups, delineating standard approaches to structural and permutation representations. A radically new presentation of the theory of varieties of lattice-ordered groups is offered.;This work is intended for pure and applied mathematicians and algebraists interested in topics such as group, order, number and lattice theory, universal algebra, and representation theory; and upper-level undergraduate and graduate students in these disciplines.;College or university bookstores may order five or more copies at a special student price which is available from Marcel Dekker Inc, upon request.

Author : Valeriĭ Matveevich Kopytov,V.M. Kopytov,Nikolai Yakovlevich Medvedev
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 53,5 Mb
Release : 1996-04-30
Category : Mathematics
ISBN : 0306110601

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Right-Ordered Groups by Valeriĭ Matveevich Kopytov,V.M. Kopytov,Nikolai Yakovlevich Medvedev Pdf

The notion of right-ordered groups is fundamental in theories of I-groups, ordered groups, torsion-free groups, and the theory of zero-divisors free rings, as well as in theoretical physics. Right-Ordered Groups is the first book to provide a systematic presentation of right-ordered group theory, describing all known and new results in the field. The volume addresses topics such as right-ordered groups and order permutation groups, the system of convex subgroups of a right-ordered group, and free products of right-ordered groups.

Author : Stuart A. Steinberg
Publisher : Springer Science & Business Media
Page : 639 pages
File Size : 54,7 Mb
Release : 2009-11-19
Category : Mathematics
ISBN : 9781441917218

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Lattice-ordered Rings and Modules by Stuart A. Steinberg Pdf

This book provides an exposition of the algebraic aspects of the theory of lattice-ordered rings and lattice-ordered modules. All of the background material on rings, modules, and lattice-ordered groups necessary to make the work self-contained and accessible to a variety of readers is included. Filling a gap in the literature, Lattice-Ordered Rings and Modules may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules. Steinberg presents the material through 800+ extensive examples of varying levels of difficulty along with numerous exercises at the end of each section. Key topics include: lattice-ordered groups, rings, and fields; archimedean $l$-groups; f-rings and larger varieties of $l$-rings; the category of f-modules; various commutativity results.

Author : Andrew Martin William Glass
Publisher : Cambridge University Press
Page : 333 pages
File Size : 45,9 Mb
Release : 1981
Category : Mathematics
ISBN : 9780521241908

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Ordered Permutation Groups by Andrew Martin William Glass Pdf

As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

Author : Andrew Martin William Glass,Holland W Charles
Publisher : Unknown
Page : 380 pages
File Size : 42,6 Mb
Release : 1989
Category : Electronic
ISBN : 3792301164

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Lattice Ordered Groups: Advances and Techniques by Andrew Martin William Glass,Holland W Charles Pdf

Author : W. Charles Holland
Publisher : CRC Press
Page : 216 pages
File Size : 50,6 Mb
Release : 2001-04-01
Category : Mathematics
ISBN : 9781482283150

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Ordered Algebraic Structures by W. Charles Holland Pdf

This book is an outcome of the conference on ordered algebraic structures held at Nanjing. It covers a range of topics: lattice theory, ordered semi groups, partially ordered groups, totally ordered groups, lattice-ordered groups, and ordered fields.

Author : Paul F. Conrad
Publisher : Unknown
Page : 326 pages
File Size : 54,5 Mb
Release : 1970
Category : Group theory
ISBN : STANFORD:36105033265476

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Lattice Ordered Groups by Paul F. Conrad Pdf

Author : W.C. Holland
Publisher : Springer Science & Business Media
Page : 252 pages
File Size : 55,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461334439

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Ordered Groups and Infinite Permutation Groups by W.C. Holland Pdf

The subjects of ordered groups and of infinite permutation groups have long en joyed a symbiotic relationship. Although the two subjects come from very different sources, they have in certain ways come together, and each has derived considerable benefit from the other. My own personal contact with this interaction began in 1961. I had done Ph. D. work on sequence convergence in totally ordered groups under the direction of Paul Conrad. In the process, I had encountered "pseudo-convergent" sequences in an ordered group G, which are like Cauchy sequences, except that the differences be tween terms of large index approach not 0 but a convex subgroup G of G. If G is normal, then such sequences are conveniently described as Cauchy sequences in the quotient ordered group GIG. If G is not normal, of course GIG has no group structure, though it is still a totally ordered set. The best that can be said is that the elements of G permute GIG in an order-preserving fashion. In independent investigations around that time, both P. Conrad and P. Cohn had showed that a group admits a total right ordering if and only if the group is a group of automor phisms of a totally ordered set. (In a right ordered group, the order is required to be preserved by all right translations, unlike a (two-sided) ordered group, where both right and left translations must preserve the order.