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Lecture Notes On Algebraic Structure Of Lattice Ordered Rings

Author: Jingjing Ma
Publisher: World Scientific
ISBN: 981457144X
Size: 39.52 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: 73.76 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: 24.61 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.

Lattice Ordered Rings And Modules

Author: Stuart A. Steinberg
Publisher: Springer Science & Business Media
ISBN: 1441917217
Size: 13.34 MB
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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.

Lectures On Algebra

Author: Shreeram Shankar Abhyankar
Publisher: World Scientific
ISBN: 9812773444
Size: 67.63 MB
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This book is a timely survey of much of the algebra developed during the last several centuries including its applications to algebraic geometry and its potential use in geometric modeling.The present volume makes an ideal textbook for an abstract algebra course, while the forthcoming sequel, Lectures on Algebra II, will serve as a textbook for a linear algebra course. The author''s fondness for algebraic geometry shows up in both volumes, and his recent preoccupation with the applications of group theory to the calculation of Galois groups is evident in the second volume which contains more local rings and more algebraic geometry. Both books are based on the author''s lectures at Purdue University over the last few years.

Lattices And Ordered Algebraic Structures

Author: T.S. Blyth
Publisher: Springer Science & Business Media
ISBN: 184628127X
Size: 65.81 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

Groups Rings And Fields

Author: David A.R. Wallace
Publisher: Springer Science & Business Media
ISBN: 1447104250
Size: 78.52 MB
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This is a basic introduction to modern algebra, providing a solid understanding of the axiomatic treatment of groups and then rings, aiming to promote a feeling for the evolutionary and historical development of the subject. It includes problems and fully worked solutions, enabling readers to master the subject rather than simply observing it.

Lectures On Finite Fields And Galois Rings

Author: Zhe-Xian Wan
Publisher: World Scientific
ISBN: 9789812385703
Size: 30.26 MB
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This is a textbook for graduate and upper level undergraduate students in mathematics, computer science, communication engineering and other fields. The explicit construction of finite fields and the computation in finite fields are emphasised. In particular, the construction of irreducible polynomials and the normal basis of finite fields are included. The essentials of Galois rings are also presented. This invaluable book has been written in a friendly style, so that lecturers can easily use it as a text and students can use it for self-study. A great number of exercises have been incorporated.

Axioms For Lattices And Boolean Algebras

Author: Ranganathan Padmanabhan
Publisher: World Scientific
ISBN: 9812834540
Size: 64.42 MB
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The importance of equational axioms emerged initially with the axiomatic approach to Boolean algebras, groups, and rings, and later in lattices. This unique research monograph systematically presents minimal equational axiom-systems for various lattice-related algebras, regardless of whether they are given in terms of ?join and meet? or other types of operations such as ternary operations. Each of the axiom-systems is coded in a handy way so that it is easy to follow the natural connection among the various axioms and to understand how to combine them to form new axiom systems. A new topic in this book is the characterization of Boolean algebras within the class of all uniquely complemented lattices. Here, the celebrated problem of E V Huntington is addressed, which ? according to G Gratzer, a leading expert in modern lattice theory ? is one of the two problems that shaped a century of research in lattice theory. Among other things, it is shown that there are infinitely many non-modular lattice identities that force a uniquely complemented lattice to be Boolean, thus providing several new axiom systems for Boolean algebras within the class of all uniquely complemented lattices. Finally, a few related lines of research are sketched, in the form of appendices, including one by Dr Willian McCune of the University of New Mexico, on applications of modern theorem-proving to the equational theory of lattices.

Grobner Bases In Ring Theory

Author: Huishi Li
Publisher: World Scientific
ISBN: 9814365149
Size: 11.35 MB
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This monograph strives to introduce a solid foundation on the usage of GrAbner bases in ring theory by focusing on noncommutative associative algebras defined by relations over a field K. It also reveals the intrinsic structural properties of GrAbner bases, presents a constructive PBW theory in a quite extensive context and, along the routes built via the PBW theory, the book demonstrates novel methods of using GrAbner bases in determining and recognizing many more structural properties of algebras, such as the GelfandOCoKirillov dimension, Noetherianity, (semi-)primeness, PI-property, finiteness of global homological dimension, Hilbert series, (non-)homogeneous p-Koszulity, PBW-deformation, and regular central extension. With a self-contained and constructive GrAbner basis theory for algebras with a skew multiplicative K-basis, numerous illuminating examples are constructed in the book for illustrating and extending the topics studied. Moreover, perspectives of further study on the topics are prompted at appropriate points. This book can be of considerable interest to researchers and graduate students in computational (computer) algebra, computational (noncommutative) algebraic geometry; especially for those working on the structure theory of rings, algebras and their modules (representations).