The Structure of Finite Algebras (Contemporary Mathematics)

Authors : David Hobby, Department of Mathematics, State University of New York New Paltz and Ralph McKenzie, Department of Mathematics, Vanderbilt University
ISBN : 0821850733
Pages : 203
Publication Date : August 1988
Publisher : American Mathematical Society

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.

View/Download The Structure of Finite Algebras (Contemporary Mathematics)