The Structure of Finite Algebras (Contemporary Mathematics)

This book begins with a straightforward and complete development of basic tame congruence theory, a topic that offers a wide variety of investigations. It then moves beyond the consideration of individual algebras to a study of locally finite varieties.

**Tag(s):**
Algebra

**Publication date**: 01 Aug 1988

**ISBN-10**:
0821850733

**ISBN-13**:
n/a

**Paperback**:
203 pages

**Views**: 15,252

The Structure of Finite Algebras (Contemporary Mathematics)

This book begins with a straightforward and complete development of basic tame congruence theory, a topic that offers a wide variety of investigations. It then moves beyond the consideration of individual algebras to a study of locally finite varieties.

Book Excerpts:

Finite algebra in this book means a finite set of elements together with a (possibly infinite) set of operations acting on this set of elements. This concept includes finite groups and rings and many other algebraic systems of interest in mathematics. Excluded are finite systems with infinitary operations, and those having "partial operations" (operations defined for some, but not all, n-tuples of elements). This book regards a locally finite variety as a class of algebras of one type, closed under the formation of homomorphic images, subalgebras, and direct products, whose finitely generated algebras are finite. The class of groups satisfying x3 = 1 is an example of a locally finite variety.

The main discovery presented in this book is that the lattice of congruences of a finite algebra determines very deeply the structure of that algebra. The authors' 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. This book uses the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well.

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 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.

Finite algebra in this book means a finite set of elements together with a (possibly infinite) set of operations acting on this set of elements. This concept includes finite groups and rings and many other algebraic systems of interest in mathematics. Excluded are finite systems with infinitary operations, and those having "partial operations" (operations defined for some, but not all, n-tuples of elements). This book regards a locally finite variety as a class of algebras of one type, closed under the formation of homomorphic images, subalgebras, and direct products, whose finitely generated algebras are finite. The class of groups satisfying x3 = 1 is an example of a locally finite variety.

The main discovery presented in this book is that the lattice of congruences of a finite algebra determines very deeply the structure of that algebra. The authors' 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. This book uses the theory to derive many new results that will be of interest not only to universal algebraists, but to other algebraists as well.

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 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.

Tweet

About The Author(s)

No information is available for this author.

No information is available for this author.

Book Categories

Computer Science
Introduction to Computer Science
Introduction to Computer Programming
Algorithms and Data Structures
Artificial Intelligence
Computer Vision
Machine Learning
Neural Networks
Game Development and Multimedia
Data Communication and Networks
Coding Theory
Computer Security
Information Security
Cryptography
Information Theory
Computer Organization and Architecture
Operating Systems
Image Processing
Parallel Computing
Concurrent Programming
Relational Database
Document-oriented Database
Data Mining
Big Data
Data Science
Digital Libraries
Compiler Design and Construction
Functional Programming
Logic Programming
Object Oriented Programming
Formal Methods
Software Engineering
Agile Software Development
Information Systems
Geographic Information System (GIS)

Mathematics
Mathematics
Algebra
Abstract Algebra
Linear Algebra
Number Theory
Numerical Methods
Precalculus
Calculus
Differential Equations
Category Theory
Proofs
Discrete Mathematics
Theory of Computation
Graph Theory
Real Analysis
Complex Analysis
Probability
Statistics
Game Theory
Queueing Theory
Operations Research
Computer Aided Mathematics

Supporting Fields
Web Design and Development
Mobile App Design and Development
System Administration
Cloud Computing
Electric Circuits
Embedded System
Signal Processing
Integration and Automation
Network Science
Project Management

Operating System
Programming/Scripting
Ada
Assembly
C / C++
Common Lisp
Forth
Java
JavaScript
Lua
Microsoft .NET
Rexx
Perl
PHP
Python
R
Rebol
Ruby
Scheme
Tcl/Tk

Miscellaneous
Sponsors