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.)?
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A good, user-friendly, modern, introductory textbook is Davey and Priestley's Introduction to Lattices and Order.
Incidentally, Gian-Carlo Rota used to say much the same thing as you, Zev: that lattice theory had been robbed of its rightful place in mathematics.
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$\begingroup$ Thanks for the recommendation! Also, I think I may have heard Rota's quote somewhere, but couldn't remember its correct attribution, or whether it was a quote, etc., hence the similar phrasing. $\endgroup$ Commented Feb 6, 2010 at 1:42
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$\begingroup$ I second this recommendation. I was just looking through this book again a few days ago and reminded how very nice it is. $\endgroup$– Steve DCommented Feb 6, 2010 at 1:47
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1$\begingroup$ If you haven't already read Rota's Notices article, here's a link to it: ams.org/notices/199711/comm-rota.pdf $\endgroup$– j.c.Commented Feb 6, 2010 at 2:15
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2$\begingroup$ Good link, jc. How's this for an opening sentence? "Never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory." Rota could certainly grab your attention. $\endgroup$ Commented Feb 6, 2010 at 2:39
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2$\begingroup$ I had the good fortune to learn lattice theory from Priestley herself and completely concur with the sentiment that there is so much more to lattice theory than is commonly thought. The book is very readable and extremely interesting. I recommend it highly. $\endgroup$ Commented Feb 6, 2010 at 20:08
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I 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 the lattice field is not my field.
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1$\begingroup$ @Leonid: I was going to post pretty much exactly what you said. $\endgroup$ Commented Feb 6, 2010 at 3:28
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2$\begingroup$ There are three editions, and they differ from one another more than one would expect. $\endgroup$– TriCommented Sep 5, 2021 at 20:22
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1$\begingroup$ @Tri, could you, please, drop more details $\endgroup$ Commented Sep 7, 2021 at 22:24
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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.
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3$\begingroup$ G. Gratzer's "General Lattice Theory" is a really thorough monograph. (Beware: it's over 600 pages long!). $\endgroup$ Commented Mar 12, 2010 at 16:15
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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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1$\begingroup$ Ross Willard's "Overview of Modern Universal Algebra" math.uwaterloo.ca/~rdwillar/documents/Publications/torino.pdf gives some answers to the question in the last paragraph above (i.e. what are some of the big theorems of universal algebra?). $\endgroup$ Commented Dec 10, 2010 at 0:44
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2$\begingroup$ Link-rot remedy for Burris and Sankappanavar: people.math.sc.edu/mcnulty/alglatvar/burrissanka.pdf. $\endgroup$– LSpiceCommented Sep 3, 2014 at 0:12
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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
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Searching online I found a couple of books:
- Steven Roman, Lattices and Ordered Sets, Springer, 2008.
- T.S. Blyth, Lattices and Ordered Algebraic Structures, Universitext, Springer, 2005.
and a set of lecture notes, available for free from the author's webpage:
J.B. Nation, Notes on Lattice Theory, revised 2017.
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$\begingroup$ I'm a fan of Nation's notes. $\endgroup$ Commented Sep 6, 2021 at 11:32
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I agree with many of the suggestions. I will also add "Algebraic Theory of Lattices" by Crawley and Dilworth, although I must admit I have not read all of it.