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Chapters: Arity, Subalgebra, Structure, Closure Operator, Post's Lattice, Basis, Ultraproduct, Tarski's High School Algebra Problem, Signature, Variety, Pseudoelementary Class, Substructure, Graph Algebra, Subdirect Irreducible, Vector Spaces Without Fields, Subdirect Product, Group With Operators, Bck Algebra, Clone, Generating Set, Term Algebra, Reduct, Quasivariety, Quasiidentity, Universal Algebraic Geometry. Source: Wikipedia. Pages: 111. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: 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. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. In database theory, structures with no functions are studied as models for relational databases, in the form of relational models. Formally, a structure can be defined as a triple consisting of a domain A, a signature , and an interpretation function I that indicates how the signature is to be interpreted on the domain. To indicate that a structure has a particular signature one can refer to it as a -structure. The domain of a structure is an arbitrary set; it is also called the underlying set of the structure, its carrier (especially in universal algebra), or its universe (especially in model theory). Very often the definition of a structure prohibits the empty domain. Sometimes the...More: http://booksllc.net/?id=4055928 ... Read more

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Chapters: Determinant, Power Set, Dimension, Isomorphism, Universal Algebra, Binomial Theorem, Algebraic Number, Identity Element, Automorphism, Union, Kernel, Congruence Relation, Operator, Algebraic Extension, Ratio, Factorization, Coefficient, Rational Root Theorem, Operand, Complement, Inverse Element, Recurrence Relation, Cyclotomic Polynomial, History of Elementary Algebra, Laws of Form, Partial Fraction, New Algebra, K-Theory, Timeline of Algebra, Derivative, Order of Operations, Ping-Pong Lemma, Grigorchuk Group, Timeline of Algebra and Geometry, Grushko Theorem, Infrastructure, Variable, Boundedly Generated Group, Completing the Square, Distributive Lattice, Association Scheme, Rank of a Group, the Compendious Book on Calculation by Completion and Balancing, Graph Dynamical System, Algebraic Function, Schur Algebra, Irreducible Polynomial, Multinomial Theorem, Variety, Filtration, Bose-mesner Algebra, Canonical Form, Permanent, Digital Root, Closed-Form Expression, Intersection, Nested Radical, List of Basic Algebra Topics, Symmetric Difference, Monomial, Ax-grothendieck Theorem, Field Arithmetic, Brahmagupta-fibonacci Identity, Cycle, Freshman's Dream, Goursat's Lemma, Transforming Polynomials, Stack, Median Algebra, Spherical Design, Degree, Algebraic Element, Haran's Diamond Theorem, Continuant, Unital, Absolutely Irreducible, Differential Equations of Addition, Conjugate, Tertiary Ideal, Constant, Reduct, Immanant of a Matrix, Distributive Homomorphism, Substitution, Polynomial Diophantine Equation, Prosolvable Group, Componendo and Dividendo, Temperley-lieb Algebra, Operation Theory, Indeterminate, Real Structure, Generalized Arithmetic Progression, Algebraic Biology, Plugging In, de Morgan Algebra, Invertible Module, Symbolic Method, Topological Module, Trinomial, Monomial Basis, Alternatization, Sylvester's Determinant Theorem, Theory of Equations, Multiplicative Digital Root, Content, Aperiodic Finite State Autom...http://booksllc.net/?id=9550030 ... Read more

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In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties below, are included among the algebraic structures by tradition. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here. ... Read more

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Algebraic methods, in particular those of universal algebra and algebraic logic, are playing an increasingly important role in computer science, especially in the areas of algebraic specification of data types, relational data types, relational database theory, logic of programmes, functional and logic programming, and semantics of programming languages. To a large extent this work has been carried forward by computer scientists independent of the very active group of mathematicians who work in universal algebra and algebraic logic. A conference was held at Iowa State University in June 1988 to bring together leading researchers from both groups to identify areas of common interest. Addresses were given by Joel Berman, H.Peter Gumm, Bjarni Jonsson, Dexter Kozen, Istvan Nemeti, Vaughan Pratt, Dana Scott and Eric Wagner. The programme also included contributed papers and a round-table discussion of the role of algebra and logic in computer science. ... Read more

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This book, which is the first systematic exposition of the algebraic approach to representations of Lie groups via representations of (or modules over) the corresponding universal enveloping algebras, turned out to be so well written that even today it remains one of the main textbooks and reference books on the subject. In 1992, Jacques Dixmier was awarded the Leroy P. Steele Prize for expository writing in mathematics. The Committee's citation mentioned Enveloping Algebras as one of Dixmier's ``extraordinary books''. Written with unique precision and elegance, the book provides the reader with insight and understanding of this very important subject. It can be an excellent textbook for a graduate course, as well as a very useful source of references in the theory of universal enveloping algebras, the area of mathematics that remains as important today as it was 20 years ago. For the 1996 printing, the author updated the status of open problems and added some relevant references. ... Read more

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This is a textbook examining the Yang-Baxter equation. The Yang-Baxter equation was presented a quarter of a century ago and became one of the main concerns of physicists and mathematicians in recent years. This book arose from lectures given by the author in an attempt to reformulate the results in the rapidly developing research works and make the materials more accessible. It explains the presentation of the Yang-Baxter equation from statistical models, and expounds systematically the meaning and solving methods for this equation. From the viewpoint of theoretical physicists it intends to develop an intuitive understanding of the fundamental knowledge on the Hopf algebras, quantization of Lie bialgebras, and the quantum enveloping algebras, but placing emphasis on the introduction of the calculation skill in terms of the physical language. ... Read more

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The book consists of the lectures presented at the NATO ASI on`Algebras and Orders' held in 1991 at the Université deMontréal. The lectures cover a broad spectrum of topics inuniversal algebra, Boolean algebras, lattices and orders, and theirlinks with graphs, relations, topology and theoretical computerscience. More specifically, the contributions deal with the followingtopics: Abstract clone theory (W. Taylor); Hyperidentities andhypervarieties (D. Schweigert); Arithmetical algebras and varieties(A. Pixley); Boolean algebras with operators (B. Jonsson); Algebraicduality (B. Davey); Model-theoretic aspects of partial algebras (P.Burmeister); Free lattices (R. Freese); Algebraic ordered sets (M.Erné); Diagrams of orders (I. Rival); Essentially minimalgroupoids (H. Machida, I.G. Rosenberg); and Formalization of predicatecalculus (I. Fleischer). Most of the papers are up-to-date surveyswritten by leading researchers, or topics that are either new or havewitnessed recent substantial progress. In most cases, the surveys arethe first available in the literature. The book is accessible tograduate students and researchers. ... Read more

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Natural duality theory is one of the major growth areas within general algebra. This text provides a short path to the forefront of research in duality theory. It presents a coherent approach to new results in the area, as well as exposing open problems.

Unary algebras play a special role throughout the text. Individual unary algebras are relatively simple and easy to work with. But as a class they have a rich and complex entanglement with dualisability. This combination of local simplicity and global complexity ensures that, for the study of natural duality theory, unary algebras are an excellent source of examples and counterexamples.

A number of results appear here for the first time. In particular, the text ends with an appendix that provides a new and definitive approach to the concept of the rank of a finite algebra and its relationship with strong dualisability.