A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - lattices seem to show up everywhere, the author or teacher says "observe that these ____ form a complete lattice" or something similar, and then moves on, never to speak of what that might imply. But, not currently knowing anything about them, I can't be sure. What would be a good place to learn about lattice theory, especially its implications for "naturally occurring" lattices (subgroups, ideals, etc.)?
George Grätzer has written a couple of well-regarded books on lattices. The wikipedia page recommends his "Lattice theory. First concepts and distributive lattices" and several others.
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.
If you want to see lattice theory in action, check out a book on Universal Algebra. Graetzer wrote such a text, so I imagine (but do not know from experience) that he will have many such examples; I cut my teeth on "Algebras, Lattices, Varieties", which has a gentle introduction to lattice theory from a universal algebraic point of view, followed by many universal algebraic results depending on that introduction. This was co-written by my advisor, Ralph McKenzie.
(Hopefully others will share examples from other fields that use lattices.)
Gerhard "Ask Me About System Design" Paseman, 2010.02.06
Searching online I found a couple of books:
- Roman - Lattices and Ordered Sets
- Blyth - Lattices and Ordered Algebraic Structures
and a set of lecture notes, available for free from the author's webpage: