Download or read book Mal cev Protomodular Homological and Semi Abelian Categories written by Francis Borceux and published by Springer Science & Business Media. This book was released on 2004-02-29 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory. The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment. The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.
Download or read book Commutator Theory for Congruence Modular Varieties written by Ralph Freese and published by CUP Archive. This book was released on 1987-08-20 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Varieties of Groups written by Hanna Neumann and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Varieties of algebras are equationally defined classes of algebras, or "primitive classes" in MAL'CEV'S terminology. They made their first explicit appearance in the 1930's, in Garrett BIRKHOFF'S paper on "The structure of abstract algebras" and B. H. NEUMANN'S paper "Identical relations in groups I". For quite some time after this, there is little published evidence that the subject remained alive. In fact, however, as part of "universal algebra", it aroused great interest amongst those who had access, directly or indirectly, to PHILIP HALL'S lectures given at Cambridge late in the 1940's. More recently, category theory has provided a general setting since varieties, suitably interpreted, are very special examples of categories. Whether their relevance to category theory goes beyond this, I do not know. And I doubt that the category theoretical approach to varieties will be more than a fringe benefit to group theory. Whether or not my doubts have substance, the present volume owes its existence not to the fact that varieties fit into a vastly more general pattern, but to the benefit group theory has derived from the classification of groups by varietal properties. It is this aspect of the study of varieties that seems to have caused its reappearance in the literature in the 1950's.
Download or read book Modes written by Anna B. Romanowska and published by World Scientific. This book was released on 2002 with total page 640 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces the theory and application of modes, covering topics in universal algebra, category theory, and modal theory, and includes exercises to illustrate concepts.
Download or read book Algebraic Systems written by Anatolij Ivanovic Mal'cev and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 331 pages. Available in PDF, EPUB and Kindle. Book excerpt: As far back as the 1920's, algebra had been accepted as the science studying the properties of sets on which there is defined a particular system of operations. However up until the forties the overwhelming majority of algebraists were investigating merely a few kinds of algebraic structures. These were primarily groups, rings and lattices. The first general theoretical work dealing with arbitrary sets with arbitrary operations is due to G. Birkhoff (1935). During these same years, A. Tarski published an important paper in which he formulated the basic prin ciples of a theory of sets equipped with a system of relations. Such sets are now called models. In contrast to algebra, model theory made abun dant use of the apparatus of mathematical logic. The possibility of making fruitful use of logic not only to study universal algebras but also the more classical parts of algebra such as group theory was dis covered by the author in 1936. During the next twenty-five years, it gradually became clear that the theory of universal algebras and model theory are very intimately related despite a certain difference in the nature of their problems. And it is therefore meaningful to speak of a single theory of algebraic systems dealing with sets on which there is defined a series of operations and relations (algebraic systems). The formal apparatus of the theory is the language of the so-called applied predicate calculus. Thus the theory can be considered to border on logic and algebra.
Download or read book The Theory of Classes of Groups written by Guo Wenbin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the characteristics of modern algebra is the development of new tools and concepts for exploring classes of algebraic systems, whereas the research on individual algebraic systems (e. g. , groups, rings, Lie algebras, etc. ) continues along traditional lines. The early work on classes of alge bras was concerned with showing that one class X of algebraic systems is actually contained in another class F. Modern research into the theory of classes was initiated in the 1930's by Birkhoff's work [1] on general varieties of algebras, and Neumann's work [1] on varieties of groups. A. I. Mal'cev made fundamental contributions to this modern development. ln his re ports [1, 3] of 1963 and 1966 to The Fourth All-Union Mathematics Con ference and to another international mathematics congress, striking the ories of classes of algebraic systems were presented. These were later included in his book [5]. International interest in the theory of formations of finite groups was aroused, and rapidly heated up, during this time, thanks to the work of Gaschiitz [8] in 1963, and the work of Carter and Hawkes [1] in 1967. The major topics considered were saturated formations, Fitting classes, and Schunck classes. A class of groups is called a formation if it is closed with respect to homomorphic images and subdirect products. A formation is called saturated provided that G E F whenever Gjip(G) E F.
Download or read book Polynomial Completeness in Algebraic Systems written by Kalle Kaarli and published by CRC Press. This book was released on 2000-07-21 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Boolean algebras have historically played a special role in the development of the theory of general or "universal" algebraic systems, providing important links between algebra and analysis, set theory, mathematical logic, and computer science. It is not surprising then that focusing on specific properties of Boolean algebras has lead to new directions in universal algebra. In the first unified study of polynomial completeness, Polynomial Completeness in Algebraic Systems focuses on and systematically extends another specific property of Boolean algebras: the property of affine completeness. The authors present full proof that all affine complete varieties are congruence distributive and that they are finitely generated if and only if they can be presented using only a finite number of basic operations. In addition to these important findings, the authors describe the different relationships between the properties of lattices of equivalence relations and the systems of functions compatible with them. An introductory chapter surveys the appropriate background material, exercises in each chapter allow readers to test their understanding, and open problems offer new research possibilities. Thus Polynomial Completeness in Algebraic Systems constitutes an accessible, coherent presentation of this rich topic valuable to both researchers and graduate students in general algebraic systems.
Download or read book The Structure of Finite Algebras written by David Charles Hobby and published by . This book was released on 1988 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The utility of congruence lattices in revealing the structure of general algebras has been recognized since Garrett Birkhoff's pioneering work in the 1930s and 1940s. However, the results presented in this book are of very recent origin: most of them were developed in 1983. The main discovery presented here is that the lattice of congruences of a finite algebra is deeply connected to the structure of that algebra. The theory reveals a sharp division of locally finite varieties of algebras into six interesting new families, each of which is characterized by the behavior of congruences in the algebras. The authors use the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well. The authors begin with a straightforward and complete development of basic tame congruence theory, a topic that offers great promise for a wide variety of investigations. They then move beyond the consideration of individual algebras to a study of locally finite varieties. A list of open problems closes the work.
Download or read book Categories Allegories written by P.J. Freyd and published by Elsevier. This book was released on 1990-11-08 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: General concepts and methods that occur throughout mathematics – and now also in theoretical computer science – are the subject of this book. It is a thorough introduction to Categories, emphasizing the geometric nature of the subject and explaining its connections to mathematical logic. The book should appeal to the inquisitive reader who has seen some basic topology and algebra and would like to learn and explore further.The first part contains a detailed treatment of the fundamentals of Geometric Logic, which combines four central ideas: natural transformations, sheaves, adjoint functors, and topoi. A special feature of the work is a general calculus of relations presented in the second part. This calculus offers another, often more amenable framework for concepts and methods discussed in part one. Some aspects of this approach find their origin in the relational calculi of Peirce and Schroeder from the last century, and in the 1940's in the work of Tarski and others on relational algebras. The representation theorems discussed are an original feature of this approach.
Download or read book A Course in Universal Algebra written by S. Burris and published by Springer. This book was released on 2011-10-21 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and a student entering the subject now will find a bewildering amount of material to digest. This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests. Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included. Chapter II develops the most general and fundamental notions of uni versal algebra-these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems. Free algebras are discussed in great detail-we use them to derive the existence of simple algebras, the rules of equational logic, and the important Mal'cev conditions. We introduce the notion of classifying a variety by properties of (the lattices of) congruences on members of the variety. Also, the center of an algebra is defined and used to characterize modules (up to polynomial equivalence). In Chapter III we show how neatly two famous results-the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's character ization of languages accepted by finite automata-can be presented using universal algebra. We predict that such "applied universal algebra" will become much more prominent.
Download or read book Universal Algebra written by George Grätzer and published by Springer Science & Business Media. This book was released on 2008-12-15 with total page 601 pages. Available in PDF, EPUB and Kindle. Book excerpt: Universal Algebra has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well selected additional bibliography of over 1250 papers and books which makes this an indispensable new edition for students, faculty, and workers in the field.
Download or read book Galois Theory Hopf Algebras and Semiabelian Categories written by George Janelidze, Bodo Pareigis, and Walter Tholen and published by American Mathematical Soc.. This book was released on with total page 588 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Complexity of Constraints written by Nadia Creignou and published by Springer. This book was released on 2008-12-23 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nowadays constraint satisfaction problems (CSPs) are ubiquitous in many different areas of computer science, from artificial intelligence and database systems to circuit design, network optimization, and theory of programming languages. Consequently, it is important to analyze and pinpoint the computational complexity of certain algorithmic tasks related to constraint satisfaction. The complexity-theoretic results of these tasks may have a direct impact on, for instance, the design and processing of database query languages, or strategies in data-mining, or the design and implementation of planners. This state-of-the-art survey contains the papers that were invited by the organizers after conclusion of an International Dagstuhl-Seminar on Complexity of Constraints, held in Dagstuhl Castle, Germany, in October 2006. A number of speakers were solicited to write surveys presenting the state of the art in their area of expertise. These contributions were peer-reviewed by experts in the field and revised before they were collated to the 9 papers of this volume. In addition, the volume contains a reprint of a survey by Kolaitis and Vardi on the logical approach to constraint satisfaction that first appeared in 'Finite Model Theory and its Applications', published by Springer in 2007.
Download or read book Algebras Lattices Varieties written by Ralph S. Freese and published by American Mathematical Society. This book was released on 2022-10-28 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.
Download or read book Galois Theory Hopf Algebras and Semiabelian Categories written by George Janelidze and published by American Mathematical Soc.. This book was released on 2004 with total page 582 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabeliancategories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers,specifically those interested in Galois theory and Hopf algebras and their categorical unification.
Download or read book The Shape of Congruence Lattices written by Keith Kearnes and published by American Mathematical Soc.. This book was released on 2013-02-26 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is concerned with the relationships between Maltsev conditions, commutator theories and the shapes of congruence lattices in varieties of algebras. The authors develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. They prove that a residually small variety that satisfies a congruence identity is congruence modular.
Download or read book The Lattice of Subquasivarieties of a Locally Finite Quasivariety written by Jennifer Hyndman and published by Springer. This book was released on 2018-08-28 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the ways in which the algebras in a locally finite quasivariety determine its lattice of subquasivarieties. The book starts with a clear and comprehensive presentation of the basic structure theory of quasivariety lattices, and then develops new methods and algorithms for their analysis. Particular attention is paid to the role of quasicritical algebras. The methods are illustrated by applying them to quasivarieties of abelian groups, modular lattices, unary algebras and pure relational structures. An appendix gives an overview of the theory of quasivarieties. Extensive references to the literature are provided throughout.
