Friday, 18 July 2008

ra.rings and algebras - Good lattice theory books?

I agree with Gerhard. Imho, "Algebras, Lattices, Varieties I" is the best book on universal algebra and lattice theory (perhaps the best math book ever ;) Ironically, it's out of print. However, Burris and Sankapanavar is also great and is free.



As far as sharing examples of the utility of lattice theory, personally, I don't know how I got through my comps in groups, rings, and fields before I learned about lattice theory. Now the only way I can remember many of the theorems is to picture the subgroup (subring, subfield) lattice!



Professor Lampe's Notes on Galois Theory and G-sets are great examples of how these subjects can be viewed abstractly from a universal algebra/lattice theory perspective. The Galois theory notes in particular distil the theory to its basic core, making it very elegant and easy to remember, and highlighting the fact that the underlying algebras need not be fields.



There is still the question of what results are truly universal algebra results, rather than old results couched in universal algebra language? That is an interesting question, and maybe should be the subject of a different mathoverflow post...



Updates: See also this post.

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