Lattices And Ordered Algebraic Structures

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Lattices And Ordered Algebraic Structures

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Author : T.S. Blyth
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-04-18

Lattices And Ordered Algebraic Structures written by T.S. Blyth and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-04-18 with Mathematics categories.

"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

Lattices And Ordered Algebraic Structures

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Author : T.S. Blyth
language : en
Publisher: Springer
Release Date : 2009-10-12

Lattices And Ordered Algebraic Structures written by T.S. Blyth and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-10-12 with Mathematics categories.

"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

Lattices And Ordered Algebraic Structures

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Author : T.S. Blyth
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-11-24

Lattices And Ordered Algebraic Structures written by T.S. Blyth and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-11-24 with Mathematics categories.

The notion of an order plays an important role ˆ not only throughout mat- maticsbutalsoinadjacentdisciplinessuchaslogicandcomputerscience. The purpose of the present text is to provide a basic introduction to the theory of ordered structures. Taken as a whole, the material is mainly designed for a postgraduate course. However, since prerequisites are minimal, selected parts of it may easily be considered suitable to broaden the horizon of the advanced undergraduate. Indeed, this has been the author’s practice over many years. A basic tool in analysis is the notion of a continuous function, namely a mapping which has the property that the inverse image of an open set is an open set. In the theory of ordered sets there is the corresponding concept of a residuated mapping, this being a mapping which has the property that the inverse image of a principal down-set is a principal down-set. It comes the- fore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present - position, the naturality of residuated mappings can perhaps best be exhibited using categorical concepts. If we regard an ordered set as a small category then an order-preserving mapping f : A? B becomes a functor. Then f is + + residuated if and only if there exists a functor f : B? A such that (f,f ) is an adjoint pair.

Lattices And Ordered Sets

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Author : Steven Roman
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-15

Lattices And Ordered Sets written by Steven Roman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-15 with Mathematics categories.

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.

Introduction To Lattices And Order

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Author : B. A. Davey
language : en
Publisher: Cambridge University Press
Release Date : 2002-04-18

Introduction To Lattices And Order written by B. A. Davey and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-04-18 with Mathematics categories.

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.

The Theory Of Lattice Ordered Groups

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Author : V.M. Kopytov
language : en
Publisher: Springer Science & Business Media
Release Date : 1994-10-31

The Theory Of Lattice Ordered Groups written by V.M. Kopytov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-10-31 with Mathematics categories.

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.

Lecture Notes On Algebraic Structure Of Lattice Ordered Rings

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Author : Jingjing Ma
language : en
Publisher: World Scientific
Release Date : 2014-03-14

Lecture Notes On Algebraic Structure Of Lattice Ordered Rings written by Jingjing Ma and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-14 with Mathematics categories.

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.

Fundamental Structures Of Algebra And Discrete Mathematics

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Author : Stephan Foldes
language : en
Publisher: John Wiley & Sons
Release Date : 2011-02-14

Fundamental Structures Of Algebra And Discrete Mathematics written by Stephan Foldes and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-02-14 with Mathematics categories.

Introduces and clarifies the basic theories of 12 structural concepts, offering a fundamental theory of groups, rings and other algebraic structures. Identifies essentials and describes interrelationships between particular theories. Selected classical theorems and results relevant to current research are proved rigorously within the theory of each structure. Throughout the text the reader is frequently prompted to perform integrated exercises of verification and to explore examples.

Lattice Ordered Groups

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Author : M.E Anderson
language : en
Publisher: Springer Science & Business Media
Release Date : 1988-01-31

Lattice Ordered Groups written by M.E Anderson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988-01-31 with Computers categories.

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].

Partially Ordered Algebraic Systems

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Author : Laszlo Fuchs
language : en
Publisher: Courier Corporation
Release Date : 2014-03-05

Partially Ordered Algebraic Systems written by Laszlo Fuchs and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-05 with Mathematics categories.

This monograph by a distinguished mathematician constitutes the first systematic summary of research concerning partially ordered groups, semigroups, rings, and fields. The high-level, self-contained treatment features numerous problems. 1963 edition.