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.)?

A good, userfriendly, modern, introductory textbook is Davey and Priestley's Introduction to Lattices and Order. Incidentally, GianCarlo Rota used to say much the same thing as you, Zev: that lattice theory had been robbed of its rightful place in mathematics. 


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 Gsets 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 cowritten 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 


I've used Garret Birkhoff book "Lattice theory". Could be a bit outdated nowadays, but it gives a deep feeling. Not so sure if it is good as lattice field is not my field. 


George Grätzer has written a couple of wellregarded books on lattices. The wikipedia page recommends his "Lattice theory. First concepts and distributive lattices" and several others. 


Searching online I found a couple of books:
and a set of lecture notes, available for free from the author's webpage: 

