Introduction to the Theory of Abstract Algebras

First things first, let’s sort out a source of possible confusion. The book under review dates to 1968, and its title might make modern readers expect its contents to range over groups, rings, fields, modules, etc., topics which today fall under the umbrella of “abstract algebra”. That is entirely not the case. A more informative title these days would be “An introduction to universal algebra”. The book consists of five chapters: basic concepts, subdirect decompositions, direct decompositions, free algebras, and varieties of algebras (and a preliminary one devoted to set theoretic prerequisites). The first chapter presents the terminology, and each of the remaining chapters revolves around one main concept, largely with one main result as an aim.

If this review was written around the time the book originally appeared, it would look something like this: “The book under review is an excellent introduction to the fundamentals of universal algebra, succinctly yet clearly presenting, in five short chapters, some of the main results in the area. The technical prerequisites are very modest, basically a bit of set theory. Any student who had finished a first course in advanced algebra and is wondering as to the nature of similarities between various constructions and structure theorems of, say, groups and rings is amply motivated to find answers in the book. The book also serves as a foundations for further reading in the area.”

Since nearly half a century had passed since the writing of the book, my review nowadays must include the following more critical addendum which is, obviously, not a criticism of the book but rather of the linear nature of the passage of time.

Universal algebra is the study of algebraic structures in a very broad sense, encompassing many different algebraic structures, and providing, through numerous powerful theorems, a great uniformity and understanding of recurring themes that the student encounters time and again when studying group theory, then ring theory, then module theory, etc. But to gain entry into universal algebra as presented in the book one must be willing to be confronted with a rather cumbersome set-theoretic formalism, a formalism that may burden the reader and knock the wind out of her sails.

The development of a categorical approach to universal algebra, in particular the concept of Lawvere theory, offer, in my view, a much more palatable journey into the realm of universal algebra, and it is related to current research. The book being what it is — an excellent text with much to gain from reading it — remains recommended, but the reader will do well to consult modern treatments as well and view, with pleasure, how the language of category theory encapsulates much detail behind intuitive concepts so that the mind is freed from the burden of counting arities and indices, to rejoice at the core matter.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.