A survey on Universal Algebra

Question

I've been searching for a good list of books and sources on Universal Algebra. Since the closest I could get from any site was this post, I decided to create a new post.

For this, I would like suggestions of books of the following three types of introductory books in universal algebra: (1) advanced books on Universal Algebra, (2) books in Model Theory, but that have a good enough amount of Universal Algebra to be on this list, (3) lecture notes on Universal Algebra or lecture notes on Model Theory that includes some topics in Universal Algebra. A little description of the book will also be great (thank you), and if you did in fact read the book, a little review would be awesome.

The idea is to let the post, so new books could always be added to the list. Edit: In one of my many searches through the internet, I found this, the best source I could find so far.

Also, one thing I am really interested in is video lectures on Universal Algebra; if someone finds some, post them here please.

Answer 1

This might be considered too broad to remain open, but I will add some fuel to the fire. These are from memory, and should be reviewed and corrected. I invite others to edit to expand upon my impressions and dim recollections.

Whitehead has the earliest text I am aware of, called Universal Algebra. This was before the modern version of the subject and would contain much found in algebra courses today. Bourbaki has a similar text with a more structured flavor, but again precedes the modern theory. Various advanced algebra texts touch on the modern subject.

Garrett Birkhoff has Lattice Theory, which is essential for understanding much of the work done in Universal Algebra in the last century that involves congruences, varieties, and many other characters of the field.

George Graetzer has one of the earliest comprehensive texts to the modern version of the field. His focus is on partial algebras, but covers much of the initial subject that was known up to the 1960's. I don't know if it has been updated. Graetzer has a modern edition of lattice theory as well, which I have not read.

Burris and Sankappannavar have an introductory book that is part of the Springer series. They are interested in Boolean powers and subjects related to interpretability. However, one can read the first three or so chapters to get a good introduction to the field.

I cut my teeth on Algebras, Lattices, Varieties, Vol. I by McKenzie, McNulty, and Taylor. Chapter 1 gives motivation and an introduction, chapter 2 covers the lattice theory material, chapter three focuses on certain classes of unwary and binary algebras, and chapter four leads to free algebras and the HSP theorem, as well as touches on clone theory and other topics. Chapter five focuses on decomposition into factors and when that is possible. I have seen manuscripts for part of volume II, but nothing in print.

Of course I have been exposed to other topics pioneered by Ralph McKenzie and others. Tame congruence theory and commutator theory are subjects developed in the 80's and 90's, and their titles can be found with a web search. Much has been done about decidability, but I don't know if books have appeared. Search for papers by Jeong, McKenzie, and Valeriote.

Edit: Horrors! Arturo Magidin reminded me of George Bergman's 245 notes, entitled An Invitation To General Algebra And Universal Constructions. I don't know how I forgot that. I must be getting stupid.

Other works have appeared on partial algebras. I believe one such author is Peter Burmeister.

It's possible that Ralph Freese or Bill Lampe have written a book on lattice theory, updating developments there.

Some members of MathOverflow have available texts. Jaroslav Jezek has his version on Universal Algebra, Mark Sapir and Olga Kharlompovich have one concerning algorithmic problems, and some of the aforementioned texts on Model Theory touch on the subject of Universal Algebra. Even some category theorists write on the subject.

I'd be interested in other texts in Universal Algebra published in this century.

Gerhard "Been Out Of The Loop" Paseman, 2019.08.08.

Answer 2

I want to highlight several of the monographs mentioned on Ralph Freese's list (linked above). General references, particularly older ones, don't really give a sense of the state of the field. Universal algebra is a much deeper subject than you would gather from seeing theorems such as the Birkhoff Variety Theorem.

One major development has been a general theory of commutators that generalizes the commutator in group theory to much broader classes of algebras. A standard reference for this is Freese and McKenzie's Commutator Theory for Congruence Modular Varieties, though much more is known now than is in the book. You can define rather broadly notions of abelian, solvable and nilpotent algebras, and use them to study the structure theory of algebras in different varieties. A monograph that marks the state of the art is Kearnes and Kiss' The Shape of Congruence Lattices, though it's not an elementary introduction.

Another major development is a theory for finite algebras, first developed in Hobby and McKenzie's The Structure of Finite Algebras. Here another new tool is developed, that of tame congruence theory. Finite simple algebras turn out to fall into five distinct families, which you can think of rought as one of: 1) unary, 2) abelian groups, 3) Boolean algebras, 4) lattices, 5) semilattices. For example, simple groups are either type 2 or type 3. (The fact that nonabelian simple groups are type 3 is implicitly the content of a theorem of Maurer and Rhodes in group theory). We can say even more, and attach to any congruence and a congruence that covers it one of the five types, so in some sense any finite algebra is built up out of pieces of only five types. A general finite group, for example, is built up only of pieces of types 2 and 3. A semigroup can have all five types, which is one reason why the theory of groups is so much nicer than the theory of semigroups.

One major recent research push which isn't really covered by Freese's list is the application of universal algebra techniques to analyze finite constraint satisfaction problems. The goal is to show that all such problems are either in P or are NP-complete. I'm not quite sure the status of this, though.

Answer 3

Since we are considering old texts, a really nice reference is the 1979 survey Equational Logic by Walter Taylor, that appeared in the Houston Journal of Mathematics in 1979. (You can check a review at the JSL.)

Another book, that I only skimmed over is Paul M. Cohn's Universal Algebra.

Finally, once you've gone through the basics, you could take a look at the monograph Finite Algebra by Clasen and Valeriote. I liked that very much.