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I think a useful combination of resources for universal algebra would ideally, when taken together:

  1. Provide ample motivation behind the various developments in the field.
  2. Either provide powerful intuitions, or avoid them altogether, in an effort to not cripple the learner's imagination.
  3. Provide a clear formal development (or, minimizes the number of "rabbits pulled out of a hat").
  4. Be self-contained.
  5. Be well written.

I look forward to your suggestions, and thank you sincerely in advance!

Rushabh Mehta
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bzm3r
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    I used Burris and Sankappanavar's "a course in universal algebra" to learn about this topic and I found very clear. – Egbert Feb 04 '14 at 21:33
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    @Egbert I wish you would write up your suggestion in an answer, so that I could upvote (and eventually accept it, after having done some reading through it myself). Perhaps you could also write a little about how you moved on from it to other resources after reading through it? – bzm3r Feb 04 '14 at 21:39

2 Answers2

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This is a good question, and Eran gives a good answer. Here is another collection of resources (which I just created in response to this question):

https://github.com/UniversalAlgebra/UAResources

Although there are many pages scattered around the interweb that collect valuable resources in universal algebra and lattice theory, it seems to me that none of them (except possibly http://universalalgebra.org) encourage contributions from all members of the math community. Consequently, we don't have one authoritative, central repository that people could point to when answering questions like this one.

The UAResources repository is just a quick first pass, but maybe it will grow with community support. One thing in particular that UAResources currently lacks is a "road map" that would directly answer to the OP's question.

Perhaps someone is willing to fill in the "Road Map" section of the README.md page at https://github.com/UniversalAlgebra/UAResources#road-map

William DeMeo
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    While I know that "thank you" comments are discouraged, in this case, I feel that I had to do so, given the huge amount of effort you put into setting up the GitHub page with links to resources. It is very much appreciated! – bzm3r Feb 05 '14 at 07:20
  • I thought I might run by a small suggestion I have for starting off the "roadmap". I had experience with Boolean algebra (in the context of correct-by-construction programs), and then because I am really an engineering student, I came across some problems that forced me to charactersize "space and form". In that context, I came across geometric algebra (born of Grassmann+Clifford+Hestenes+others). I was so amazed by how various axiomatizations of algebraic operations could model something physical...it was something that I only realized later I had seen before with Boolean algebra. – bzm3r Feb 05 '14 at 08:08
  • A bit later, when working on a problem dealing with supremums/infimums etc. I bumped across "Universal Algebra" and I saw glimpses of names I knew (Boole, Grassmann)...then I found this paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.39&rep=rep1&type=pdf...I didn't really understand much of it. – bzm3r Feb 05 '14 at 08:11
  • I wondered though, if universal algebra could help me understand what it is about algebraic structure that is "analogous" with physical stuff...and I wonder if there is a paper somewhere that explores this route of motivation. I wonder if it would be an interesting "0" in the roadmap. Curious to hear what you have to say! – bzm3r Feb 05 '14 at 08:12
  • @twirlobite Re your first comment: No thanks necessary. As I suggested, the repository is something that should have existed already, and it took very little time to set up. I hope people decide to contribute to it, but to be honest I'm not very optimistic. Perhaps as people get more comfortable with things like Git and GitHub, it will have a better chance of success. – William DeMeo Feb 06 '14 at 20:42
  • @twirlobite Unfortunately, I don't have an answer to your question right now. I learned some geometric algebra and computer vision as a grad student. I suggested to my mentors that this could make a good seminar topic, but they steered me away from it, and instead we began a seminar in residuated structures and logic. Since I now do "pure" math, I feel they guided me in the right direction, but your question about universal algebra helping one understand how algebraic structures model physical things is a good one. I'll give it some thought and maybe post a comment in the "road map." – William DeMeo Feb 06 '14 at 21:04
  • I am not well connected with academia at all, nor do I know much pure math, so I don't think there is much I can do to contribute to the repository at the moment. What I will try to do however is keep you up-to-date with any discoveries I make along the line of exploration I outlined above, and I will continue to eagerly look forward to anything you have to add with enthusiasm. – bzm3r Feb 06 '14 at 22:52
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I must admit that I am still in the process of learning universal algebra, so my answer may not be as good as that of a researcher in the field, for example.

My recommendation is for Clifford Bergman's Universal Algebra. It is very clearly written and organized. The book has a large bibliography and gives good introductions to many of the current research areas.

Other good resources are the book by Burris and Sankappanavar and the book by McKenzie, McNulty, and Taylor. Burris and Sanka's book is available for free online and Mckenzie is one of the premier researchers in universal algebra.

Once you have learned enough of the basics you could probably begin reading one of the following:

Hobby and McKenzie's book on tame congruence theory

McKenzie and Freese's book on commutator theory

Clark and Davey's book on duality theory

Eran
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