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Author: Vi_acheslav Nikolaevich Sali_ Publisher: American Mathematical Soc. ISBN: 9780821898178 Category : Mathematics Languages : en Pages : 132
Book Description
The class of uniquely complemented lattices properly contains all Boolean lattices. However, no explicit example of a non-Boolean lattice of this class has been found. In addition, the question of whether this class contains any complete non-Boolean lattices remains unanswered. This book focuses on these classical problems of lattice theory and the various attempts to solve them. Requiring no specialized knowledge, the book is directed at researchers and students interested in general algebra and mathematical logic.
Author: Vi_acheslav Nikolaevich Sali_ Publisher: American Mathematical Soc. ISBN: 9780821898178 Category : Mathematics Languages : en Pages : 132
Book Description
The class of uniquely complemented lattices properly contains all Boolean lattices. However, no explicit example of a non-Boolean lattice of this class has been found. In addition, the question of whether this class contains any complete non-Boolean lattices remains unanswered. This book focuses on these classical problems of lattice theory and the various attempts to solve them. Requiring no specialized knowledge, the book is directed at researchers and students interested in general algebra and mathematical logic.
Author: Vi︠a︡cheslav Nikolaevich Saliĭ Publisher: ISBN: 9781470444839 Category : Lattice theory Languages : en Pages :
Book Description
The class of uniquely complemented lattices properly contains all Boolean lattices. However, no explicit example of a non-Boolean lattice of this class has been found. In addition, the question of whether this class contains any complete non-Boolean lattices remains unanswered. This book focuses on these classical problems of lattice theory and the various attempts to solve them. Requiring no specialized knowledge, the book is directed at researchers and students interested in general algebra and mathematical logic.
Author: B. A. Davey Publisher: Cambridge University Press ISBN: 9780521784511 Category : Mathematics Languages : en Pages : 316
Book Description
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.
Author: Steven Roman Publisher: Springer Science & Business Media ISBN: 0387789014 Category : Mathematics Languages : en Pages : 307
Book Description
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.
Author: Ralph N. McKenzie Publisher: American Mathematical Society ISBN: 1470442957 Category : Mathematics Languages : en Pages : 386
Book Description
This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text.
Author: Garrett Birkhoff Publisher: American Mathematical Soc. ISBN: 0821810251 Category : Mathematics Languages : en Pages : 434
Book Description
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
Author: T.S. Blyth Publisher: Springer Science & Business Media ISBN: 184628127X Category : Mathematics Languages : en Pages : 311
Book Description
"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
Author: Peter Eklund Publisher: Springer ISBN: 3540246517 Category : Mathematics Languages : en Pages : 418
Book Description
This volume contains the Proceedings of ICFCA 2004, the 2nd International Conference on Formal Concept Analysis. The ICFCA conference series aims to be the premier forum for the publication of advances in applied lattice and order theory and in particular scienti?c advances related to formal concept analysis. Formal concept analysis emerged in the 1980s from e?orts to restructure lattice theory to promote better communication between lattice theorists and potentialusersoflatticetheory.Sincethen,the?eldhasdevelopedintoagrowing research area in its own right with a thriving theoretical community and an increasing number of applications in data and knowledge processing including data visualization, information retrieval, machine learning, data analysis and knowledge management. In terms of theory, formal concept analysis has been extended into attribute exploration, Boolean judgment, contextual logic and so on to create a powerful general framework for knowledge representation and reasoning. This conference aims to unify theoretical and applied practitioners who use formal concept an- ysis, drawing on the ?elds of mathematics, computer and library sciences and software engineering. The theme of the 2004 conference was ‘Concept Lattices” to acknowledge the colloquial term used for the line diagrams that appear in almost every paper in this volume. ICFCA 2004 included tutorial sessions, demonstrating the practical bene?ts of formal concept analysis, and highlighted developments in the foundational theory and standards. The conference showcased the increasing variety of formal concept analysis software and included eight invited lectures from distinguished speakersinthe?eld.Sevenoftheeightinvitedspeakerssubmittedaccompanying papers and these were reviewed and appear in this volume.
Author: Thomas Donnellan Publisher: Elsevier ISBN: 1483147495 Category : Mathematics Languages : en Pages : 296
Book Description
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.
Author: Vi_acheslav Nikolaevich Sali_
Author: Vi_acheslav Nikolaevich Sali_
Author: Vi︠a︡cheslav Nikolaevich Saliĭ
Author: B. A. Davey
Author: Steven Roman
Author: Manfred Stern
Author: Ralph N. McKenzie
Author: Garrett Birkhoff
Author: T.S. Blyth
Author: Peter Eklund
Author: Thomas Donnellan