Introduction to Relation Algebras

Relation Algebras

Author: Steven Givant

Publisher: Springer

ISBN: 3319652354

Category: Mathematics

Page: 572

View: 373

The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly suited to independent study, and provide an unparalleled opportunity to learn from one of the leading authorities in the field. Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community.
Posted in Mathematics

Advanced Topics in Relation Algebras

Relation Algebras

Author: Steven Givant

Publisher: Springer

ISBN: 3319659456

Category: Mathematics

Page: 605

View: 1651

The second volume of a pair that charts relation algebras from novice to expert level, this text brings the well-grounded reader to the frontiers of research. Building on the foundations established in the preceding Introduction to Relation Algebras, this volume advances the reader into the deeper mathematical results of the past few decades. Such material offers an ideal preparation for research in relation algebras and Boolean algebras with operators. Arranged in a modular fashion, this text offers the opportunity to explore any of several areas in detail; topics include canonical extensions, completions, representations, varieties, and atom structures. Each chapter offers a complete account of one such avenue of development, including a historical section and substantial number of exercises. The clarity of exposition and comprehensive nature of each module make this an ideal text for the independent reader entering the field, while researchers will value it as a reference for years to come. Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community. Note that this volume contains numerous, essential references to the previous volume, Introduction to Relation Algebras. The reader is strongly encouraged to secure at least electronic access to the first book in order to make use of the second.
Posted in Mathematics

Relation Algebras by Games

Author: Robin Hirsch,Ian Hodkinson

Publisher: Gulf Professional Publishing

ISBN: 9780444509321

Category: Mathematics

Page: 691

View: 374

In part 2, games are introduced, and used to axiomatise various classes of algebras. Part 3 discusses approximations to representability, using bases, relation algebra reducts, and relativised representations. Part 4 presents some constructions of relation algebras, including Monk algebras and the 'rainbow construction', and uses them to show that various classes of representable algebras are non-finitely axiomatisable or even non-elementary. Part 5 shows that the representability problem for finite relation algebras is undecidable, and then in contrast proves some finite base property results. Part 6 contains a condensed summary of the book, and a list of problems. There are more than 400 exercises. P The book is generally self-contained on relation algebras and on games, and introductory text is scattered throughout. Some familiarity with elementary aspects of first-order logic and set theory is assumed, though many of the definitions are given.-
Posted in Mathematics

Relation Algebras

Author: Steven Givant

Publisher: Springer

ISBN: 9783319685809

Category: Mathematics

Page: N.A

View: 4578

Collecting, curating, and illuminating over 75 years of progress since Tarski's seminal work in 1941, this textbook in two volumes offers a landmark, unified treatment of the increasingly relevant field of relation algebras. Clear and insightful prose guides the reader through material previously only available in scattered, highly-technical journal articles. Students and experts alike will appreciate the work as both a textbook and invaluable reference for the community. This set charts relation algebras from novice to expert level. The first volume, Introduction to Relation Algebras, offers a comprehensive grounding for readers new to the topic. The second, Advanced Topics in Relation Algebras, build on this foundation and advances the reader into the deeper mathematical results of the past few decades. Such material offers an ideal preparation for research in relation algebras and Boolean algebras with operators. Note that the second volume contains numerous, essential references to the first. Readers of the advanced material are encouraged to purchase the pair as a set, as access to the first book is necessary to make use of the second.
Posted in Mathematics

An Introduction to Abstract Algebra

Author: F. M. Hall

Publisher: CUP Archive

ISBN: 9780521298612

Category: Mathematics

Page: 316

View: 7884

This two-volume course on abstract algebra provides a broad introduction to the subject for those with no previous knowledge of it but who are well grounded in ordinary algebraic techniques. It starts from the beginning, leading up to fresh ideas gradually and in a fairly elementary manner, and moving from discussion of particular (concrete) cases to abstract ideas and methods. It thus avoids the common practice of presenting the reader with a mass of ideas at the beginning, which he is only later able to relate to his previous mathematical experience. The work contains many concrete examples of algebraic structures. Each chapter contains a few worked examples for the student - these are divided into straightforward and more advanced categories. Answers are provided. From general sets, Volume 1 leads on to discuss special sets of the integers, other number sets, residues, polynomials and vectors. A chapter on mappings is followed by a detailed study of the fundamental laws of algebra, and an account of the theory of groups which takes the idea of subgroups as far as Langrange's theorem. Some improvements in exposition found desirable by users of the book have been incorporated into the second edition and the opportunity has also been taken to correct a number of errors.
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Algebraic Logic

Author: H. Andréka,James Donald Monk,I. Németi

Publisher: North Holland

ISBN: N.A

Category: Mathematics

Page: 746

View: 2566

This volume is not restricted to papers presented at the 1988 Colloquium, but instead aims to provide the reader with a (relatively) coherent reading on Algebraic Logic, with an emphasis on current research. To help the non-specialist reader, the book contains an introduction to cylindric and relation algebras by Roger D. Maddux and an introduction to Boolean Algebras by Bjarni Joacute;nsson.
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Decision Problems for Equational Theories of Relation Algebras

Author: H. Andréka,Steven R. Givant,I. Németi

Publisher: American Mathematical Soc.

ISBN: 0821805959

Category: Mathematics

Page: 126

View: 3645

This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.
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From Peirce to Skolem

A Neglected Chapter in the History of Logic

Author: Geraldine Brady

Publisher: Elsevier

ISBN: 9780080532028

Category: Computers

Page: 480

View: 8838

This book is an account of the important influence on the development of mathematical logic of Charles S. Peirce and his student O.H. Mitchell, through the work of Ernst Schröder, Leopold Löwenheim, and Thoralf Skolem. As far as we know, this book is the first work delineating this line of influence on modern mathematical logic.
Posted in Computers

Introduction to Abstract Algebra, Third Edition

Author: T.A. Whitelaw

Publisher: CRC Press

ISBN: 9780751401479

Category: Mathematics

Page: 256

View: 9257

The first and second editions of this successful textbook have been highly praised for their lucid and detailed coverage of abstract algebra. In this third edition, the author has carefully revised and extended his treatment, particularly the material on rings and fields, to provide an even more satisfying first course in abstract algebra.
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Introduction to Abstract Algebra

Author: Jonathan D. H. Smith

Publisher: CRC Press

ISBN: 9781420063721

Category: Mathematics

Page: 344

View: 3119

Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles. A Quick Introduction to Algebra The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level. The Basics of Abstract Algebra for a First-Semester Course Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange’s theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley’s theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields. Advanced Topics for a Second-Semester Course The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups.
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Algebraic Theories

A Categorical Introduction to General Algebra

Author: J. Adámek,J. Rosický,E. M. Vitale

Publisher: Cambridge University Press

ISBN: 1139491881

Category: Mathematics

Page: N.A

View: 3499

Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.
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Handbook of Categorical Algebra: Volume 1, Basic Category Theory

Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 9780521441780

Category: Mathematics

Page: 345

View: 5654

First of a 3-volume work giving a detailed account of what should be known by all working in, or using category theory. Volume 1 covers basic concepts.
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An Introduction to K-Theory for C*-Algebras

Author: M. Rørdam,Flemming Larsen,N. Laustsen

Publisher: Cambridge University Press

ISBN: 9780521789448

Category: Mathematics

Page: 242

View: 6640

This book provides a very elementary introduction to K-theory for C*-algebras, and is ideal for beginning graduate students.
Posted in Mathematics

Encyclopaedia of Mathematics, Supplement III

Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

ISBN: 0306483734

Category: Mathematics

Page: 557

View: 1157

This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.
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Introduction to Linear Algebra, 2nd edition

Author: T.A. Whitelaw

Publisher: CRC Press

ISBN: 9780751401592

Category: Mathematics

Page: 288

View: 4463

This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra.
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From Kant to Hilbert Volume 1

A Source Book in the Foundations of Mathematics

Author: William Bragg Ewald

Publisher: OUP Oxford

ISBN: 9780191523090

Category: Mathematics

Page: 678

View: 4883

Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics—algebra, geometry, number theory, analysis, logic and set theory—with narratives to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics.
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An Invitation to C*-Algebras

Author: W. Arveson

Publisher: Springer Science & Business Media

ISBN: 9780387901763

Category: Mathematics

Page: 108

View: 8922

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.
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Relational Mathematics

Author: Gunther Schmidt

Publisher: Cambridge University Press

ISBN: 0521762685

Category: Computers

Page: 567

View: 4262

A modern, comprehensive 2010 overview providing an easy introduction for applied scientists who are not versed in mathematics.
Posted in Computers

Introduction to Octonion and Other Non-Associative Algebras in Physics

Author: Susumo Okubo

Publisher: Cambridge University Press

ISBN: 9780521472159

Category: Science

Page: 136

View: 8321

In this book, the author applies non-associative algebras to physics. Okubo covers topics ranging from algebras of observables in quantum mechanics and angular momentum and octonions to division algebra, triple-linear products and YangSHBaxter equations. He also discusses the non-associative gauge theoretic reformulation of Einstein's general relativity theory. Much of the material found in this volume is not available in other works. The book will therefore be of great interest to graduate students and research scientists in physics and mathematics.
Posted in Science

Introduction to Non-linear Algebra

Author: Valeri? Valer?evich Dolotin,A. Morozov,Al?bert Dmitrievich Morozov

Publisher: World Scientific

ISBN: 9812708006

Category: Mathematics

Page: 269

View: 5600

Literaturverz. S. 267 - 269
Posted in Mathematics