# Introduction to the Theory of Abstract Algebras

## Richard S Pierce

- 160 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Introduction to the Theory of Abstract Algebras

## Richard S Pierce

## About This Book

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.

## Frequently asked questions

## Information

*]*

**1***Basic Concepts*

**1. Algebras and Relational Systems**

*A*be a set, and suppose that

*Ļ*is an ordinal number. A

*Ļ-ary partial operation on A*is a partial mapping of

*A*to

^{Ļ}*A.*That is,

*F*is a

*Ļ*-ary partial operation if

*F*maps

*D*ā

*A*to

^{Ļ}*A.*If

*F*) =

*A*then

^{Ļ},*F*is called a

*Ļ*-

*ary operation on A.*

*Ļ*= 0. A

*zero-ary partial operation*is either the empty set or it is a mapping from

*A*

^{0}

*=*{Ćø}

*=*{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.

*Ļ*= 1. A

*unary partial operation*on

*A*is essentially a mapping from a subset of

*A*to

*A,*since

*A*

^{1}can be identified with

*A*by the correspondence

*a*ā

*a*(0).

*Ļ*= 2. A

*binary partial operation*on

*A*corresponds to a mapping from a subset

*A*Ć

*A*to

*A*when the natural identification

*a*ā

*a*(0),

*a*(1)

*A*

^{2}and

*A*Ć

*A.*Examples of such binary operations are common. For instance, the multiplication and addition operations in a ring are binary. Frequently, composition...