An invitation to general algebra and universal constructions
George M. Bergman (Author of An Invitation to General Algebra and Universal Constructions)
Andre Henriques - Lie algebras and their representations
An Invitation to General Algebra and Universal Constructions (Universitext)
Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra plus two from topology for perspective , the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book. The author takes care in writing full proofs throughout the book and he shows also ways of possible applications.
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Universal algebra sometimes called general algebra is the field of mathematics that studies algebraic structures themselves, not examples "models" of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. In universal algebra, an algebra or algebraic structure is a set A together with a collection of operations on A. An n - ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation or nullary operation can be represented simply as an element of A , or a constant , often denoted by a letter like a. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f x , y , z or f x 1 , After the operations have been specified, the nature of the algebra is further defined by axioms , which in universal algebra often take the form of identities , or equational laws.