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Universal Algebra: Fundamentals and Selected Topics

Clifford Bergman
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Pure and Applied Mathematics
[Reviewed by
Michael Berg
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In another age, when I was at UCLA, I worked for a while as a teaching assistant for the late Alfred Horn, who was a unique figure in my panoply of professors due to the fact that he specialized in universal algebra. This was, as far as I was concerned, something truly exotic: there were a number of algebraists on the faculty, including a lattice theorist and a finite group theorist; there were algebraic geometers who knew Homeric amounts of algebra, particularly commutative algebra (but they generally didn’t care very much about, say, Sylow theory); and there were a decent number of algebraic number theorists. There was a quadratic former; a specialist in Lie groups, Lie algebras, and representation theory; and an authority on Steinberg groups (because he was Steinberg himself). In other words, just about all flavors of algebra were multiply represented, but in universal algebra there was only one player, viz. Professor Horn.

Little did I know at that time that what had drawn me in the direction of algebra in the first place, specifically the ineffable characteristic of “structure,” was specifically proper to this broadly titled subject, so that Professor Horn, quiet and unassuming as he was, could really claim legitimate membership in a number of different cliques in the department, all doing very exciting things.

In fact, it was actually the case that my own mathematical upbringing had included a pretty big dose of universal algebra, if only sub rosa, seeing that UCLA sported a number of quasi-experimental courses at that time, geared toward introducing fledgling mathematics students to, yes, structure, but eschewing the phrase “universal algebra” in their course descriptions. And it all worked quite well: even before taking group theory and linear algebra I already knew about quotient constructions, for example, and I recall fondly my first introduction to “abstract algebra” proper, in yet another course in my freshman year, where the emphasis throughout the course was on, again, structure.

The point of this reminiscence is to illustrate what I believe is an unsurpassable reason for introducing a pervasive universal algebra perspective into the entire algebra curriculum, from the earlier undergraduate stage onward and throughout what follows: the student (or learner) is equipped from the outset with an ability to absorb varying but related ideas in algebra (and its abutting areas) as “birds of a feather.” It’s a pedagogical coup to be able to deal early on with, for instance, the rank-nullity theorem of linear algebra and the first isomorphism theorem (of Emmy Noether) in group theory as kindred things, and then to come across the same business later under the heading “quotient constructions” in half a dozen other places ranging from sheaves to group schemes. After all analogy is one of the main themes in grasping and doing mathematics, and algebra is a perfect setting for it all.

Thus, as far as I am concerned, the book under review, by Clifford Bergman, is most welcome: we need more of this sort of thing, both for potential universal algebraists and for people like me: fellow travelers to some degree, or mathematicians who both use and thoroughly adore algebra and its structural qualities, and find themselves growing more appreciative of this architectural elegance as they evolve in their work and studies.

The solid core of the book’s subject is presented in Bergman’s Chapter 3, “The Nuts and Bolts of Universal Algebra,” where one encounters the themes of isomorphism theorems, direct and subdirect products, and (tantalizingly) “varieties and other classes of algebras.” Here the word “variety” is of a different variety than the variety of variety of algebraic geometry: if we’re given a class K of “similar” algebras, we say K is a variety if it contains all homomorphic images of its members, all isomorphic images of subalgebras of its members, and all isomorphic copies of direct products of its members. We see immediately that this is pretty sophisticated stuff, even if the tools in this tool box look familiar: the prevailing goal of the subject is rather different than what one would encounter in, say, group theory proper (as algebra par excellence).

Indeed, the whole affair starts off with algebras and lattices in the first two chapters, and then, after the divide provided by Chapter 3, it gets particularly sporty: clones, terms, and equational classes, then on to all sorts of esoterica about varieties in the indicated sense. Thus, as we progress through the work we see that universal algebra truly possesses a flavor all its own and is a very viable autonomous branch of algebra (with mathematico-logical overtones), and allows a good deal of creative work and research within its perhaps loosely defined borders. Of course it cannot be otherwise.

The subtitle to the book under review is “Fundamentals and Selected Topics”; as I already indicated, the third chapter is the divide between these two themes. The book is correspondingly written in an accessible and pedagogically considerate manner: it is clearly written and pleasant to read, it is not improperly fast-paced (quite the contrary), and the author provides motivation as well as examples and exercises galore. At first glance it looks to me like the exercises are well-structured and should do the job of bringing the student or reader along at a decent pace from ignorance (modulo some earlier exposure to such things as vector spaces, groups, rings, fields, &c.) to both an appreciation for the subject and some facility with it. It’s definitely an area worth pursuing for a graduate student with the right disposition.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

More about subs, homs, and prods
Generating subalgebras
Congruences and quotient algebras

Ordered sets
Distributive and modular lattices
Complete lattices
Closure operators and algebraic lattices
Galois connections
Ideals in lattices

The Nuts and Bolts of Universal Algebra
The isomorphism theorems
Direct products
Subdirect products
Case studies
Varieties and other classes of algebras

Clones, Terms, and Equational Classes
Invariant relations
Terms and free algebras
Identities and Birkhoff’s theorem
The lattice of subvarieties
Equational theories and fully invariant congruences
Maltsev conditions

Congruence Distributive Varieties
Ultrafilters and ultraproducts
Jónsson’s lemma
Model theory
Finitely based and nonfinitely based algebras
Definable principal (sub)congruences

Arithmetical Varieties
Large clones
How rare are primal algebras?

Maltsev Varieties
Directly representable varieties
The centralizer congruence
Abelian varieties
Directly representable varieties revisited
Minimal varieties
Functionally complete algebras

Finite Algebras and Locally Finite Varieties
Minimal algebras
Localization and induced algebras
Centralizers again!