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Lecture Notes On Algebraic Structure Of Lattice Ordered Rings

Author: Jingjing Ma
Publisher: World Scientific
ISBN: 981457144X
Size: 22.59 MB
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Algebraic Structure of Lattice-Ordered Rings presents an introduction to the theory of lattice-ordered rings and some new developments in this area in the last 10-15 years. It aims to provide the reader with a good foundation in the subject, as well as some new research ideas and topic in the field. This book may be used as a textbook for graduate and advanced undergraduate students who have completed an abstract algebra course including general topics on group, ring, module, and field. It is also suitable for readers with some background in abstract algebra and are interested in lattice-ordered rings to use as a self-study book. The book is largely self-contained, except in a few places, and contains about 200 exercises to assist the reader to better understand the text and practice some ideas. Contents:Introduction to Ordered Algebraic Systems:LatticesLattice-Ordered Groups and Vector LatticesLattice-Ordered Rings and AlgebrasLattice-Ordered Algebras with a d-Basis:Examples and Basic PropertiesStructure TheoremsPositive Derivations on ℓ-Rings:Examples and Basic Propertiesƒ-Ring and Its GeneralizationsMatrix ℓ-RingsKernel of a Positive DerivationSome Topics on Lattice-Ordered Rings:Recognition of Matrix ℓ-Rings with the Entrywise OrderPositive CyclesNonzero ƒ-Eelements in ℓ-RingsQuotient Rings of Lattice-Ordered Ore DomainsMatrix ℓ-Algebras Over Totally Ordered Integral Domainsd-Elements That are Not PositiveLattice-Ordered Triangular Matrix Algebrasℓ-Ideals of ℓ-Unital Lattice-Ordered Rings:Maximal ℓ-Idealsℓ-Ideals in commutative ℓ-Unital ℓ-Rings Readership: Graduate students in algebra and number theory. Key Features:The book includes new material such as positive derivations on lattice-ordered rings, lattice-ordered triangular matrix algebrasMore details are provided in proofs of the results in the book for beginners to understand the textThe book presents new research ideas, methods and topics suitable to advanced undergraduate students and master studentsKeywords:Lattice-ordered Ring;Lattice-ordered Algebra

Ordered Algebraic Structures

Author: Jorge Martínez
Publisher: Springer Science & Business Media
ISBN: 9401117233
Size: 77.97 MB
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This volume contains a selection of papers presented at the 1991 Conrad Conference, held in Gainesville, Florida, USA, in December, 1991. Together, these give an overview of some recent advances in the area of ordered algebraic structures. The first part of the book is devoted to ordered permutation groups and universal, as well as model-theoretic, aspects. The second part deals with material variously connected to general topology and functional analysis. Collectively, the contents of the book demonstrate the wide applicability of order-theoretic methods, and how ordered algebraic structures have connections with many research disciplines. For researchers and graduate students whose work involves ordered algebraic structures.

Ordered Algebraic Structures

Author: W.C. Holland
Publisher: Springer Science & Business Media
ISBN: 9401156409
Size: 44.74 MB
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The conference on Ordered Algebraic Structures held in Curat;ao, from the 26th of June through the 30th of June, 1995, at the Avila Beach Hotel, marked the eighth year of ac tivities by the Caribbean Mathematics Foundation (abbr. CMF), which was the principal sponsor of this conference. CMF was inaugurated in 1988 with a conference on Ordered Algebraic Structures. During the years between these two conferences the field has changed sufficiently, both from my point of view and, I believe, that of my co-organizer, W. Charles Holland, to make one wonder about the label "Ordered Algebraic Structures" itself. We recognized this from the start, and right away this conference carried a subtitle, or, if one prefers, an agenda: we concentrated on the one hand, on traditional themes in the theory of ordered groups, including model-theoretic aspects, and, on the other hand, on matters in which topology (more precisely C(X)-style topology) and category theory would play a prominent role. Plainly, ordered algebra has many faces, and it is becoming increas ingly difficult to organize an intimate conference, such as the ones encouraged in the series sponsored by CMF, in this area on a broad set of themes. These proceedings reflect, accurately we think, the spirit of the conferees, but it is not a faithful record of the papers presented at the conference.

Lattices And Ordered Algebraic Structures

Author: T.S. Blyth
Publisher: Springer Science & Business Media
ISBN: 184628127X
Size: 64.95 MB
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"The text can serve as an introduction to fundamentals in the respective areas from a residuated-maps perspective and with an eye on coordinatization. The historical notes that are interspersed are also worth mentioning....The exposition is thorough and all proofs that the reviewer checked were highly polished....Overall, the book is a well-done introduction from a distinct point of view and with exposure to the author’s research expertise." --MATHEMATICAL REVIEWS

An Invitation To General Algebra And Universal Constructions

Author: George M. Bergman
Publisher: Springer
ISBN: 3319114786
Size: 69.94 MB
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Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.

Zeta Functions Of Groups And Rings

Author: Marcus du Sautoy
Publisher: Springer Science & Business Media
ISBN: 354074701X
Size: 60.60 MB
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Thestudyofthesubgroupgrowthofin?nitegroupsisanareaofmathematical research that has grown rapidly since its inception at the Groups St. Andrews conferencein1985.Ithasbecomearichtheoryrequiringtoolsfromandhaving applications to many areas of group theory. Indeed, much of this progress is chronicled by Lubotzky and Segal within their book [42]. However, one area within this study has grown explosively in the last few years. This is the study of the zeta functions of groups with polynomial s- groupgrowth,inparticularfortorsion-free?nitely-generatednilpotentgroups. These zeta functions were introduced in [32], and other key papers in the - velopment of this subject include [10, 17], with [19, 23, 15] as well as [42] presenting surveys of the area. The purpose of this book is to bring into print signi?cant and as yet unpublished work from three areas of the theory of zeta functions of groups. First, there are now numerous calculations of zeta functions of groups by doctoralstudentsofthe?rstauthorwhichareyettobemadeintoprintedform outside their theses. These explicit calculations provide evidence in favour of conjectures, or indeed can form inspiration and evidence for new conjectures. We record these zeta functions in Chap.2. In particular, we document the functional equations frequently satis?ed by the local factors. Explaining this phenomenon is, according to the ?rst author and Segal [23], “one of the most intriguing open problems in the area”.

Lectures In General Algebra

Author: A. G. Kurosh
Publisher: Elsevier
ISBN: 1483149579
Size: 62.56 MB
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Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University. The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the convenience of constructing a single theory from the results of group experiments and ring experiments which are known to follow simple corollaries. The text also presents algebraic structures that are not of binary nature. From this parallelism arise other concepts, such as that of the lattices, complete lattices, and modular lattices. The book then proves the Schmidt-Ore theorem, and also describes linear algebra, as well as the Birkhoff-Witt theorem on Lie algebras. The text also addresses ordered groups, the Archimedean groups and rings, and Albert's theorem on normed algebras. This book can prove useful for algebra students and for professors of algebra and advanced mathematicians.

Theory Of Lattice Ordered Groups

Author: Michael Darnel
Publisher: CRC Press
ISBN: 9780824793265
Size: 69.82 MB
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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.

Partially Ordered Groups

Author: Andrew Martin William Glass
Publisher: World Scientific
ISBN: 9789810234935
Size: 77.59 MB
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"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