Introduction to the Theory of Abstract Algebras

Richard S Pierce

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486799421
• Topic: Matematica
• Subtopic: Algebra astratta

Intended for beginning graduate-level courses, this text introduces various aspects of the theory of abstract algebra. The book is also suitable as independent reading for interested students at that level as well as a primary source for a one-semester course that an instructor may supplement to expand to a full year. Author Richard S. Pierce, a Professor of Mathematics at Seattle's University of Washington, places considerable emphasis on applications of the theory and focuses particularly on lattice theory.
After a preliminary review of set theory, the treatment presents the basic definitions of the theory of abstract algebras. Each of the next four chapters focuses on a major theme of universal algebra: subdirect decompositions, direct decompositions, free algebras, and varieties of algebras. Problems and a Bibliography supplement the text.

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Basic Concepts

1. Algebras and Relational Systems

The purpose of this section is to introduce the principal object of interest in this book, the notion of an abstract algebra.

1.1 DEFINITION. Let A be a set, and suppose that τ is an ordinal number. A τ-ary partial operation on A is a partial mapping of A^τ to A. That is, F is a τ-ary partial operation if F maps D ⊆ A^τ to A. If

(F) = A^τ, then F is called a τ-ary operation on A.

1.2 EXAMPLES. (a) τ = 0. A zero-ary partial operation is either the empty set or it is a mapping from A⁰ = {ø} = {0} into A. In the latter case, it is an operation. Thus, the effect of a zero-ary operation on A is to specify an element of A, and it is common practice to denote a zero-ary operation by listing the element that it selects.

(b) τ = 1. A unary partial operation on A is essentially a mapping from a subset of A to A, since A¹ can be identified with A by the correspondence a → a(0).

(c) τ = 2. A binary partial operation on A corresponds to a mapping from a subset A × A to A when the natural identification a ↔

a(0), a(1)

is made between A² and A × A. Examples of such binary operations are common. For instance, the multiplication and addition operations in a ring are binary. Frequently, composition...