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Lattice Concepts Of Module Theory
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Book Synopsis Lattice Concepts of Module Theory by : Grigore Calugareanu
Download or read book Lattice Concepts of Module Theory written by Grigore Calugareanu and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc ture of its submodule lattice. Citing Louis H. Rowen "this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists". Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module theoretic results generalized in (modular, and possibly compactly gen erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con cepts (e. g.
Book Synopsis Lattice Theory: Foundation by : George Grätzer
Download or read book Lattice Theory: Foundation written by George Grätzer and published by Springer Science & Business Media. This book was released on 2011-02-14 with total page 639 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so Lattice Theory: Foundation focuses on introducing the field, laying the foundation for special topics and applications. Lattice Theory: Foundation, based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Almost 40 “diamond sections’’, many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. “Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Grätzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication.” Bulletin of the American Mathematical Society “Grätzer’s book General Lattice Theory has become the lattice theorist’s bible.” Mathematical Reviews
Book Synopsis Lattice Theory by : Garrett Birkhoff
Download or read book Lattice Theory written by Garrett Birkhoff and published by American Mathematical Soc.. This book was released on 1940-12-31 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its original publication in 1940, this book has been revised and modernized several times, most notably in 1948 (second edition) and in 1967 (third edition). The material is organized into four main parts: general notions and concepts of lattice theory (Chapters I-V), universal algebra (Chapters VI-VII), applications of lattice theory to various areas of mathematics (Chapters VIII-XII), and mathematical structures that can be developed using lattices (Chapters XIII-XVII). At the end of the book there is a list of 166 unsolved problems in lattice theory, many of which still remain open. It is excellent reading, and ... the best place to start when one wishes to explore some portion of lattice theory or to appreciate the general flavor of the field. --Bulletin of the AMS
Book Synopsis Lattice Theory by : George A. Gratzer
Download or read book Lattice Theory written by George A. Gratzer and published by . This book was released on with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis General Lattice Theory by : George Grätzer
Download or read book General Lattice Theory written by George Grätzer and published by Springer Science & Business Media. This book was released on 2002-11-21 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Grätzer’s 'General Lattice Theory' has become the lattice theorist’s bible. Now we have the second edition, in which the old testament is augmented by a new testament. The new testament gospel is provided by leading and acknowledged experts in their fields. This is an excellent and engaging second edition that will long remain a standard reference." --MATHEMATICAL REVIEWS
Book Synopsis Trends in Lattice Theory by : Garrett Birkhoff
Download or read book Trends in Lattice Theory written by Garrett Birkhoff and published by . This book was released on 1970 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Classes of Modules written by John Dauns and published by Chapman and Hall/CRC. This book was released on 2006-06-19 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Because traditional ring theory places restrictive hypotheses on all submodules of a module, its results apply only to small classes of already well understood examples. Often, modules with infinite Goldie dimension have finite-type dimension, making them amenable to use with type dimension, but not Goldie dimension. By working with natural classes and type submodules (TS), Classes of Modules develops the foundations and tools for the next generation of ring and module theory. It shows how to achieve positive results by placing restrictive hypotheses on a small subset of the complement submodules, Furthermore, it explains the existence of various direct sum decompositions merely as special cases of type direct sum decompositions. Carefully developing the foundations of the subject, the authors begin by providing background on the terminology and introducing the different module classes. The modules classes consist of torsion, torsion-free, s[M], natural, and prenatural. They expand the discussion by exploring advanced theorems and new classes, such as new chain conditions, TS-module theory, and the lattice of prenatural classes of right R-modules, which contains many of the previously used lattices of module classes. The book finishes with a study of the Boolean ideal lattice of a ring. Through the novel concepts presented, Classes of Modules provides a new, unexplored direction to take in ring and module theory.
Download or read book Lattice Theory written by George Grätzer and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Lattice Theory by : Thomas Donnellan
Download or read book Lattice Theory written by Thomas Donnellan and published by Elsevier. This book was released on 2014-05-16 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lattice Theory presents an elementary account of a significant branch of contemporary mathematics concerning lattice theory. This book discusses the unusual features, which include the presentation and exploitation of partitions of a finite set. Organized into six chapters, this book begins with an overview of the concept of several topics, including sets in general, the relations and operations, the relation of equivalence, and the relation of congruence. This text then defines the relation of partial order and then partially ordered sets, including chains. Other chapters examine the properties of meet and join and explain dimensional considerations. This book discusses as well certain relations between individual elements of a lattice, between subsets of a lattice, and between lattices themselves. The final chapter deals with distributive lattices and explores the complements in distributive lattices. This book is a valuable resource for college and university students of mathematics, logic, and such technologies as communications engineering.
Book Synopsis Introduction to Lattices and Order by : B. A. Davey
Download or read book Introduction to Lattices and Order written by B. A. Davey and published by Cambridge University Press. This book was released on 2002-04-18 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.
Book Synopsis General Lattice Theory by : George A. Gratzer
Download or read book General Lattice Theory written by George A. Gratzer and published by Birkhauser. This book was released on 1998 with total page 696 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this present edition, the work has been significantly updated and expanded. It contains an extensive new bibliography of 530 items and has been supplemented by eight appendices authored by an exceptional group of experts. The first appendix, written by the author, briefly reviews developments in lattice theory, specifically, the major results of the last 20 years and solutions of the problems proposed in the first edition. The other subjects concern distributive lattices and duality (Brian A. Davey and Hilary A. Priestley), continuous geometries (Friedrich Wehrung), projective lattice geometries (Marcus Greferath and Stefan E. Schmidt), varieties (Peter Jipsen and Henry Rose), free lattices (Ralph Freese), formal concept analysis (Bernhard Ganter and Rudolf Wille), and congruence lattices (Thomas Schmidt in collaboration with the author).
Book Synopsis Lattice Theory and the Theory of Semi-simple Modules and Rings by : Rosemary Grace Bonyun
Download or read book Lattice Theory and the Theory of Semi-simple Modules and Rings written by Rosemary Grace Bonyun and published by . This book was released on 1961 with total page 68 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis General Lattice Theory by : George Grätzer
Download or read book General Lattice Theory written by George Grätzer and published by Birkhäuser. This book was released on 2002-12-05 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Grätzer’s 'General Lattice Theory' has become the lattice theorist’s bible. Now we have the second edition, in which the old testament is augmented by a new testament. The new testament gospel is provided by leading and acknowledged experts in their fields. This is an excellent and engaging second edition that will long remain a standard reference." --MATHEMATICAL REVIEWS
Book Synopsis Lattices and Ordered Sets by : Steven Roman
Download or read book Lattices and Ordered Sets written by Steven Roman and published by Springer Science & Business Media. This book was released on 2008-12-15 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to be a thorough introduction to the subject of order and lattices, with an emphasis on the latter. It can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area. This is a book on pure mathematics: I do not discuss the applications of lattice theory to physics, computer science or other disciplines. Lattice theory began in the early 1890s, when Richard Dedekind wanted to know the answer to the following question: Given three subgroups EF , and G of an abelian group K, what is the largest number of distinct subgroups that can be formed using these subgroups and the operations of intersection and sum (join), as in E?FßÐE?FÑ?GßE?ÐF?GÑ and so on? In lattice-theoretic terms, this is the number of elements in the relatively free modular lattice on three generators. Dedekind [15] answered this question (the answer is #)) and wrote two papers on the subject of lattice theory, but then the subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Grätzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann.
Book Synopsis Lattice Theory: Special Topics and Applications by : George Grätzer
Download or read book Lattice Theory: Special Topics and Applications written by George Grätzer and published by Springer. This book was released on 2014-08-27 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: George Grätzer's Lattice Theory: Foundation is his third book on lattice theory (General Lattice Theory, 1978, second edition, 1998). In 2009, Grätzer considered updating the second edition to reflect some exciting and deep developments. He soon realized that to lay the foundation, to survey the contemporary field, to pose research problems, would require more than one volume and more than one person. So Lattice Theory: Foundation provided the foundation. Now we complete this project with Lattice Theory: Special Topics and Applications, written by a distinguished group of experts, to cover some of the vast areas not in Foundation. This first volume is divided into three parts. Part I. Topology and Lattices includes two chapters by Klaus Keimel, Jimmie Lawson and Ales Pultr, Jiri Sichler. Part II. Special Classes of Finite Lattices comprises four chapters by Gabor Czedli, George Grätzer and Joseph P. S. Kung. Part III. Congruence Lattices of Infinite Lattices and Beyond includes four chapters by Friedrich Wehrung and George Grätzer.
Book Synopsis Module Theory, Extending Modules and Generalizations by : Adnan Tercan
Download or read book Module Theory, Extending Modules and Generalizations written by Adnan Tercan and published by Birkhäuser. This book was released on 2016-05-13 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main focus of this monograph is to offer a comprehensive presentation of known and new results on various generalizations of CS-modules and CS-rings. Extending (or CS) modules are generalizations of injective (and also semisimple or uniform) modules. While the theory of CS-modules is well documented in monographs and textbooks, results on generalized forms of the CS property as well as dual notions are far less present in the literature. With their work the authors provide a solid background to module theory, accessible to anyone familiar with basic abstract algebra. The focus of the book is on direct sums of CS-modules and classes of modules related to CS-modules, such as relative (injective) ejective modules, (quasi) continuous modules, and lifting modules. In particular, matrix CS-rings are studied and clear proofs of fundamental decomposition results on CS-modules over commutative domains are given, thus complementing existing monographs in this area. Open problems round out the work and establish the basis for further developments in the field. The main text is complemented by a wealth of examples and exercises.
Book Synopsis Foundations of Module and Ring Theory by : Robert Wisbauer
Download or read book Foundations of Module and Ring Theory written by Robert Wisbauer and published by Routledge. This book was released on 2018-05-11 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.
