# Which texts do you recommend to study universal algebra and lattice theory?

As I'm planning to study some algebraic logic (a lot of!), I found that some knowledge of universal algebra, lattice theory and boolean algebras is a must.

I wonder if you have any recommendation to study from to be well-prepared to study algebraic logic.

Some of the recommendations I got are:

$1$-Universal Algebra by George Graetzer. And any book on lattices by George Graetzer.

$2$- Lectures on boolean algebra by Halmos or his new text which is co-authored with Givant.

$3$- Algebraic methods in philosophical logic by Dunn and Hardegree which will gather all the stuff of lattices, universal algebra and boolean algebrs together.

Those are what I got, What do you recommend and why? Any other good books?

Should I re-ask the same question of philosophy forum?

• Universal algebra and lattice theory seems like a somewhat strange juxtaposition: lattices are just a particular case of universal algebras, which one is not necessarily going to study at the same time as universal algebra. George Bergman (at Berkeley) has a free set of course notes for universal algebra on his website which is excellent and balances the exposition between more set-centric and more categorial. Commented Oct 26, 2014 at 23:08
• There's a reason there are four different tags about lattices. You should be more specific here. Commented Oct 27, 2014 at 14:13
• @KevinCarlson, Ok, my point was that, what I knew is knowing lattices is a must in learning algebraic logic. sometimes,we need some examples in hands besides the general theory. Do you think that I don't need lattices to learn algebraic logic?
– FNH
Commented Oct 27, 2014 at 15:37
• @KevinCarlson Oops, I didn't notice that you had already recommended Bergman's book before I wrote my answer. But the lattices / universal algebra juxtaposition isn't strange at all. On the contrary, much of universal algebra is organized around the observation that a surprising amount of information about a variety of algebras is encoded in its lattices of congruences and subalgebras. Commented Oct 27, 2014 at 21:50
• @KevinCarlson Lattices are usually studied alongside universal algebra because of the fundamental role they play. Besides the examples Alex mentions, there's the lattice of varieties or equational theories. So, while it's true that lattices are "a particular case of universal algebras," they play a more central role than algebras of other types. Commented Jan 22, 2015 at 15:32

You should take a look at George Bergman's book "An Invitation to General Algebra and Universal Constructions". It's also free online, and it's a great introduction to the subject - it works through many concrete examples in Part I before getting to the abstract theory in Part II.

It doesn't contain much lattice theory, though.

• Do you think that this book fits my needs? ( I need universal algebra to learn algebraic logic)
– FNH
Commented Oct 29, 2014 at 0:47
• The book is certainly not designed with algebraic logic in mind, but I would recommend it to anyone who needed to learn universal algebra for any reason. Commented Nov 3, 2014 at 18:46

Imho, the best modern treatment of universal algebra is

Universal Algebra: Fundamentals and Selected Topics, by Cliff Bergman.

Algebras, Lattices, Varieties, by McKenzie, McNulty, Taylor is a classic and also excellent. Unfortunately, it is out of print, but most university libraries have a copy.

As mentioned by others, Burris and Sankapannavar is good (and free!). While it gives a very nice treatment of most of the basics, it was written a long time ago, and is starting to show its age. For example, there is little or no clone theory, and no mention of tame congruence theory (which was not invented until 1983). Fortunately, Cliff's new book has excellent coverage of both clones and tct.

Another list of resources for learning about universal algebra can be found here.

Unfortunately, none of the resources I've mentioned have very much to say about category theory and its relation to universal algebra and lattice theory (although, there is a very brief section in Algebras, Lattices, Varieties presenting categories from a universal algebra perspective).

Finally, since you are just getting your feet wet, I'll mention a couple more introductory references that I came across only recently (so don't know them well, but they look very interesting!) They are by well known/established authors:

Post-Modern Algebra, by Jonathan Smith and Anna Romanowska

I would like to mention the little known (but excellent) book by W. Wechler, Universal Algebra for Computer Scientists, Springer- Verlag, Berlin (1992). SBN: 978-3-642-76773-9 (Print) 978-3-642-76771-5 (Online).

It covers equational theories in great detail but also treats topics that are hardly found elsewhere, like multi-sorted algebras or ordered algebras.

Contents of the book.

1 Preliminaries

Basic Notions (Sets, Algebras Generation, Structural Induction, Algebraic Recursion and Deductive Systems)

Relations (Regular Operations, Equivalence Relations, Partial Orders, Terminating Relations, Well-quasi-Orders, Cofinality, Multiset Ordering and Polynomial Ordering).

Trees (Trees and Well-Founded Partially Ordered Sets, Labelled Trees, $\omega$-Complete Posets and Fixpoint Theorem, Free $\omega$-Completion).

2 Reductions

Word Problem (Confluence Method, Word Problem for Congruences) Reduction Systems (Abstract Reduction Systems, Term Rewriting Systems, Termination).

3 Universal Algebra

Basic Constructions (Subalgebras and Generation, Images and Presentation, Direct Products and Subdirect Decompositions, Reduced Products and Ultraproducts).

Equationally Defined Classes of Algebras (Equations, Free Algebras, Varieties, Equational Theories, Term Rewriting as an Algorithmic Tool for Equational Theories).

Implicationally Defined Classes of Algebras (Implications, Finitary Implications and Universal Horn Clauses, Sur-Reflections, Sur-Reflective Classes, Semivarieties and Quasivarieties).

Implicational Theories, Universal Horn Theories, Conditional Equational Theories and Conditional Term Rewriting.

4 Applications

Algebraic Specification of Abstract Data Types (Many-Sorted Algebras, Initial Semantics of Equational Specifications, Operational Semantics).

Algebraic Semantics of Recursive Program Schemes (Ordered Algebras, Strict Ordered Algebras, w-Complete Ordered Algebras, Recursive Program Schemes.

Appendix 1 : Sets and Classes.

Appendix 2 : Ordered Algebras as First-Order Structures.

Dunn and Hardegree's Algebraic methods in philosophical logic (2001) is no doubt a great book and very easy to cope with too. It focuses primarily on Algebraic Logic, yet the book's handling of universal algebra is interesting as well.

Another recommendation is the out-of-print A Course in Universal Algebra (1981) by S. Burris and H. P. Sankappanavarin, which is freely available online in its new version and with some corrections by the authors themselves. You can find it for download here.

• being foucused on algebraic algebra is an advantage because I'm studying universal algebra to help in in algebraic logic.
– FNH
Commented Oct 27, 2014 at 15:31
• For the free book after being corrected, Do you recommend it or not?
– FNH
Commented Oct 27, 2014 at 15:35