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This work is a revised version of the author's Ph.D. thesis written under the supervision of J. Lepowsky at Rutgers University in 1983. ... Read more

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Several of the contributions to this volume bring forward many mutually beneficial interactions and connections between the three domains of the title. Developing them was the main purpose of the NATO ASI summerschool held in Montreal in 2003. Although some connections, for example between semigroups and automata, were known for a long time, developing them and surveying them in one volume is novel and hopefully stimulating for the future. Another aspect is the emphasis on the structural theory of automata that studies ways to construct big automata from small ones. The volume also has contributions on top current research or surveys in the three domains. One contribution even links clones of universal algebra with the computational complexity of computer science. Three contributions introduce the reader to research in the former East block. ... Read more

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Motivated by applications in theoretical computer science, the theory of finite semigroups has emerged in recent years as an autonomous area of mathematics. It fruitfully combines methods, ideas and constructions from algebra, combinatorics, logic and topology. In simple terms, the theory aims at a classification of finite semigroups in certain classes called "pseudovarieties". The classifying characteristics have both structural and syntactical aspects, the general connection between them being part of universal algebra. Besides providing a foundational study of the theory in the setting approach to finite semigroups. This involves studying (relatively) free and profinite free semigroups and their presentations. The techniques used are illustrated in a systematic study of various operators on pseudovarieties of semigroups. ... Read more

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This volume is devoted to the development of an algebraic modelof databases. The first chapter presents a general introduction. Thefollowing sixteen chapters are divided into three main parts. Part Ideals with various aspects of universal algebra. The chapters of PartI discuss topics such as sets, algebras and models, fundamentalstructures, categories, the category of sets, topoi, fuzzy sets,varieties of algebras, axiomatic classes, category algebra andalgebraic theories. Part II deals with different approaches to the algebraization ofpredicate calculus. This material is intended to be applied chiefly todatabases, although some discussion of pure algebraic applications isalso given. Discussed here are topics such as Boolean algebras andpropositional calculus, Halmos algebras and predicate calculus,connections with model theory, and the categorial approach toalgebraic logic. Part III is concerned specifically with the algebraic model ofdatabases, which considers the database as an algebraic structure.Topics dealt with in this part are the algebraic aspects of databases,their equivalence and restructuring, symmetries and the Galois theoryof databases, and constructions in database theory. The volume closeswith a discussion and conclusions, and an extensive bibliography.For mathematicians, computer scientists and database engineers, withan interest in applications of algebra and logic. ... Read more

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During the last few decades the ideas, methods, and results ofthe theory of Boolean algebras have played an increasing role invarious branches of mathematics and cybernetics.
This monograph is devoted to the fundamentals of the theory of Booleanconstructions in universal algebra. Also considered are the problemsof presenting different varieties of universal algebra with theseconstructions, and applications for investigating the spectra andskeletons of varieties of universal algebras.
For researchers whose work involves universal algebra and logic.
... Read more

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Alfred North Whitehead (1861-1947) was equally celebrated as a mathematician, a philosopher and a physicist. He collaborated with his former student Bertrand Russell on the first edition of Principia Mathematica (published in three volumes between 1910 and 1913), and after several years teaching and writing on physics and the philosophy of science at University College London and Imperial College, was invited to Harvard to teach philosophy and the theory of education. A Treatise on Universal Algebra was published in 1898, and was intended to be the first of two volumes, though the second (which was to cover quaternions, matrices and the general theory of linear algebras) was never published. This book discusses the general principles of the subject and covers the topics of the algebra of symbolic logic and of Grassmann's calculus of extension. ... Read more

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Symbols, Impossible Numbers, and Geometric Entanglements is the first history of the development and reception of algebra in early modern England and Scotland.Not primarily a technical history, this book analyzes the struggles of a dozen British thinkers to come to terms with early modern algebra, its symbolical style, and negative and imaginary numbers.Professor Pycior uncovers these thinkers as a "test-group" for the symbolic reasoning that would radically change not only mathematics but also logic, philosophy, and language studies.The book also shows how pedagogical and religious concerns shaped the British debate over the relative merits of algebra and geometry.The first book to position algebra firmly in the Scientific Revolution and pursue Newton the algebraist, it highlights Newton's role in completing the evolution of algebra from an esoteric subject into a major focus of British mathematics.Other thinkers covered include Oughtred, Harriot, Wallis, Hobbes, Barrow, Berkeley, and MacLaurin. ... Read more

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Symbols, Impossible Numbers, and Geometric Entanglements
The author begins her introduction with the words: "As a history of algebra in England and Scotland from the early seventeenth century to the mid-eighteenth century, this book considers not only technical algebra but also the personal, philosophical, religious, and institutional factors that affected the introduction, elaboration, and reception of the subject." (p. 1)

The following algebra texts are discussed in this book:
Girolamo Cardano - The Great Art (Ars magna) - 1545
Francois Viete - Introduction to the Analytic Art - 1591
William Oughtred - The Key of the Mathematicks - 1631, 1647
Thomas Harriot - Praxis (Artis analyticae praxis) - 1631
Rene Descartes - Geometrie - 1637
Johann Rahn w/John Pell - Algebra - 1659, 1668
John Kersey - Algebra - 1673
John Wallis - Treatise of Algebra - 1685
Isaac Newton - Universal Arithmetick - 1707
Nicholas Saunderson - Elements of Algebra - 1740
Colin MacLaurin - Treatise of Algebra - 1748

The author also discusses views concerning algebra or mathematics by Thomas Hobbes, Isaac Barrow, and George Berkeley. The main concerns and disagreements seem to have been around the status (epistemological, ontological, or otherwise) of negative and imaginary numbers, as well as the relative standings of geometry and algebra within the realm of mathematics.

A reader wishing to know how algebra was presented within these various texts - in other words, what the algebra textbooks of the period looked like, what notation was used, how proofs were presented, what topics were included, and so forth - will need to look elsewhere. The author's main concern seems to be the controversies surrounding negative and imaginary numbers and the struggle of algebra to find respectability in the face of geometry.

The following books are available from amazon.

Cardano - The Rules of Algebra: (Ars Magna)

Viete- The Analytic Art

Thomas Harriot - Artis Analyticae Praxis: An English Translation with Commentary

Descartes - Geometry

For a look at what algebra was going to become in England during the early nineteenth century, see
Thomas Peacock's 1830 Treatise on Algebra -

Treatise on Algebra, Volume I: Arithmetical Algebra

A Treatise on Algebra, Volume II: On Symbolical Algebra, and Its Applications to the Geometry of Position

A century later, in 1930,B. L. Van Der Waerden published the enormously influential:

Algebra: Volume I

Algebra, Volume II

And then in 1941, Garrett Birkhoff and Saunders Mac Lane published:

A Survey of Modern Algebra ... Read more

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Function Algebras on Finite Sets gives a broad introduction to the subject, leading up to the cutting edge of research. The general concepts of the Universal Algebra are given in the first part of the book, to familiarize the reader from the very beginning on with the algebraic side of function algebras. The second part covers the following topics: Galois-connection between function algebras and relation algebras, completeness criterions, and clone theory.

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The utility of congruence lattices in revealing the structure ofgeneral algebras has been recognized since Garrett Birkhoff'spioneering work in the 1930s and 1940s. However, the resultspresented in this book are of very recent origin: most of themwere developed in 1983. The main discovery presented here isthat the lattice of congruences of a finite algebra is deeplyconnected to the structure of that algebra. The theory reveals asharp division of locally finite varieties of algebras into sixinteresting new families, each of which is characterized by thebehavior of congruences in the algebras. The authors use thetheory to derive many new results that will be of interest notonly to universal algebraists, but to other algebraists as well.

The authors begin with a straightforward and complete developmentof basic tame congruence theory, a topic that offers greatpromise for a wide variety of investigations. They then movebeyond the consideration of individual algebras to a study oflocally finite varieties. A list of open problems closes thework. ... Read more

Customer Reviews (1)

A well writen text for a graduate level course in Finite Algebra
Page 7 of Chapter 4 seems very interesting to me.
For me it would have been more helpful if it contained a bridge to Lie Algebras and Galois Field Algebra, but it seems enough that it
introduces directed graph notation for Lattices and sorts out different types of minimal Algebras.
Not really for the faint of heart or those with a small amount of theoretical mathematical background.
Intense study will be necessary to completely master this material. ... Read more

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The book is a fairly complete and up-to-date survey of projectivity and its generalizations in the class of Boolean algebras. Although algebra adds its own methods and questions, many of the results presented were first proved by topologists in the more general setting of (not necessarily zero-dimensional) compact spaces. An appendix demonstrates the application of advanced set-theoretic methods to the field. The intended readers are Boolean and universal algebraists. The book will also be useful for general topologists wanting to learn about kappa-metrizable spaces and related classes. The text is practically self-contained but assumes experience with the basic concepts and techniques of Boolean algebras. ... Read more

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The theory of hyperidentities generalises the equational theory of universal algebras and is applicable in several fields of science, especially in computer sciences. This book presents the theory of hyperidentities and its relation to clone identities.
The basic concept of a hypersubstitution is used to introduce the monoid of hypersubstitutions, hyperidentities, M-hyperidentities, solid, and M-solid varieties. Hypersubstitutions lead to conjugate pairs of additive closure operators. In the book the theory of such pairs of closure operations is developed. Along with this general theory, these concepts are also examined for specific varieties of algebras, especially for varieties of semigroups.
This work integrates into a coherent framework and notation many results scattered throughout the literature over the last eighteen years. In addition, the book contains some applications of hyperidentities to the functional completeness problem in multiple-valued logic. The general theory ... Read more

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"Here is the story of algebra."With this deceptively simple introduction, we begin our journey.Flanked by formulae, shadowed by roots and radicals, but escorted by an expert who navigates unerringly on our behalf, we are guaranteed safe passage through even the most treacherous mathematical terrain.

Our first encounter with algebraic arithmetic takes us back thirty-eight centuries to the time of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah.Moving deftly from Abel’s proof to the higher levels of abstraction developed by Galois, we are eventually introduced to what algebraists have been focusing on during the last century.

As we travel the ages, it becomes apparent that the invention of algebra was more than the start of a specific discipline of mathematics – it was also the birth of a new way of thinking that clarified both basic numeric concepts as well as our perception of the world around us.Algebraists broke new ground when they discarded the simple search for solutions to equations and concentrated instead on abstract groups.This dramatic shift in thinking revolutionized mathematics.

Written for those among us who are unencumbered by a fear of formulae, Unknown Quantity delivers on its promise to present a history of algebra.Astonishing in its bold presentation of the math and graced with narrative authority, our journey through the world of algebra is at once intellectually satisfying and pleasantly challenging. ... Read more

Customer Reviews (38)

excellent work in need of a layman's touch
This is allegedly written for the educated layman but I know few who could grasp (or care about) field vs group theory, rings, tensors or manifolds much less "movements" like "Intuitionism" and "Constructivism". The author thankfully does not attempt a deep inquiry into every crevice of modern math but instead presents a breezy if sometimes obtuse overview.

Even mathematicians can be informed here, particularly on the personalities.The author attempts to stay within the bounds of algebra while explaining that various disciplines necessarily cross lines with geometry, topology other algebras.The author thankfully includes illustrations although many more are required.Many times, it was next to impossible trying to picture the prose.Then again, much of the latter part of the book is conceptual without basis in reality - parallel lines that meet, many dimensions (presented in three, of course), complex numbers, vectors, etc. Many times, the creation of the mathematical concept (matrix, field permutation, etc) is as astounding as the principles by which they are solved. My Grade - B+

A nice book on the history of algebra
This book is a relatively easy to read history of algebra, togehter with short introductions to some topics in algebra that a layman might never have encountered (or needs to review).

The author gives good concrete examples of concepts such as groups, rings and fields, so that one can get an idea of what these ideas mean and why they are important.So I think it's a good introduction to some of abstract algebra; however, to understand that subject well would require more in-depth study than one could get from this book.

The narratives about prominent mathematicians are interesting and well-written too. (They also provide a sometimes-needed break from reading the more technical material.) The author also does a good job of showing the historical context for the development of certain ideas in mathematics.

Toward the end of the book, I found myself sometimes losing track of what was going on - the concepts get more and more abstract, and the explanations seem to get shorter.

In summary: this book has enough technical material so that one not familiar with the subject can learn something.I think much of it would be understandable to a motivated reader who didn't have any math background beyond what is taught in high-school algebra.

Abstraction not quite brought down to earth
Don't get me wrong, Derbyshire makes a lot of VERY abstract ideas a lot more accessible to the non-specialist reader than he/she should reasonably expect.Even so, he takes a lot more for granted this time round than he did in his last book (on prime numbers and the Riemann Hypothesis).By the end of that book, I felt that I really understood (albeit in a not especially technical way) a very difficult area of mathematical thought.Granted, UQ has as it's focus the entire mathematical branch of algebra over its 3,000 year history.I should not have expected a thorough explication of mathematical details along the way.As a survey, UQ certainly whetted my appetite to further read and explore in the field of algebra.And it was packed with fascinating information.Less one star only because JD had set such a high bar with his last book.This one doesn't quite reach it.

Group theory for beginners
This is simply brilliant.However, it is also different from John Derbyshire's other book "Prime Obsession".It is more abstract and I myself had to read in a second time to understand the concepts fully.But what a reward!Once the reader comprehends his deliberation on groups and invariance, he or she can enbark on other journeys in understanding the nature of physics (and by corollary the universe itself).So in summary, it is as clear as it can be, and as engaging as Derbyshire has been.

Good explanations + history
The historical attributions balance & complement the algebraic examples beautifully.Fun to read. ... Read more

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High Quality Content by WIKIPEDIA articles! In universal algebra and in model theory, a structure consists of a set along with a collection of finitary functions and relations which are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields, vector spaces and lattices. The term (universal) algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. ... Read more