A Course in Universal Algebra

Authors: Stanley Burris, Department of Pure Mathematics, University of Waterloo and H.P. Sankappanavar, Department of Mathematics, State University of New York
Publisher : Springer-Verlag
Publication Date : 1981

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Book Excerpts:

Universal algebra has enjoyed a particularly explosive growth in the last twenty years (ed: this book was published in 1981), and a student entering the subject now will find a bewildering amount of material to digest.

This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests.

Chapter I contains a brief but substantial introduction to lattices. and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included.

Chapter II develops the most general and fundamental notions of universal algebra these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems.

In Chapter III this book shows how neatly two famous results the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's characterization of languages accepted by finite automata can be presented using universal algebra.

Chapter IV starts with a careful development of Boolean algebras, including Stone duality, which is subsequently used in our study of Boolean sheaf representations; however, the cumbersome formulation of general sheaf theory has been replaced by the considerably simpler definition of a Boolean product.

The final chapter gives the reader a leisurely introduction to some basic concepts, tools, and results of model theory. In particular, this book uses the ultraproduct construction to derive the compactness theorem and to prove fundamental preservation theorems. Principal congruence formulas are a favorite model-theoretic tool of universal algebraists, and this book uses them in the study of the sizes of subdirectly irreducible algebras. Next this book proves three general results on the existence of a finite basis for an equational theory. The last topic is semantic embeddings, a popular technique for proving undecidability results. This technique is essentially algebraic in nature, requiring no familiarity whatsoever with the theory of algorithms. (The study of decidability has given surprisingly deep insight into the limitations of Boolean product representations.)

At the end of several sections the reader will find selected references to source material plus state of the art texts or papers relevant to that section, and at the end of the book one finds a brief survey of recent developments and several outstanding problems.

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