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I own a small book on Lattice theory published by Dover. Unfortunately, since I bought it almost a year ago, I have gotten nowhere in its study.

What I am asking for are freely available papers that use lattice theory in a substantial way to prove theorems about mathematical objects I am already familiar with and interested in: groups, rings, modules, topological vector spaces, logic, $\dots$ to help me get excited about lattices, and potentially give me a reason to study them for their own sake. Short articles ($\sim10$ pages) are especially appreciated (but not necessary), as are those articles that helped you become interested in lattice theory and convinced you of their utility.

I know this is potentially difficult, as I imagine most 'interesting' applications of lattice theory require advanced material on lattices, but I am willing to take certain theorems on faith, and have several lecture notes on lattice theory saved on my hard drive for reference.

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Flagging the mods is the best way to convert a question to CW. As long as you have 10 reputation, you can convert any of your answers to CW. – robjohn Jan 6 '13 at 19:44
@Potato It's very enlightening to have interesting applications of an abstract topic, and it's what drives the development of the abstract theory in the first place. My question is akin to "what are some interesting applications of the fundamental group", it helps you get motivated to study an object which otherwise could remain very obscure to you. Also, contrary to you and goat cheese, I don't have an aversion to lattice theory, I just want to know what it's for and how people have gained insight into familiar topics through it. – Olivier Bégassat Jan 6 '13 at 20:32
Lattices are everywhere: for example, given any algebraic structure, the set of its substructures forms a complete lattice under inclusion. But maybe you'll have better luck if your broaden your scope to cover order theory in general. – Zhen Lin Jan 6 '13 at 21:16
@ZhenLin What does the lattice structure on the set of substructures tell one? Added order theory as a tag. – Olivier Bégassat Jan 6 '13 at 21:30

If combinatorics counts as an application for you, you can take a look at Chapter 3 of Stanley's Enumerative Combinatorics.

Formal concept analysis is another example: .

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Try reading Gian-Carlo Rota's The Many Lives of Lattice Theory. This is how he starts off:

Never in the history of mathematics has a mathematical theory been the object of such vociferous vituperation as lattice theory.

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I came to lattices by trying to understand hierarchical structures as they occur in mathematics. By this I mean to say that the typical hierarchies are they occur for classes of mathematical objects are lattices. There are hierarchical structures which would fail to be lattices, take directed graphs as an example.

I'm also interested in lattice theory, because idempotent commutative semigroups are equivalent to semi-lattices. The idempotent elements of an inverse semigroup form an inverse sub-semigroup, which is useful in the analysis of a given inverse semigroups.

But why do I think that inverse semigroups are interesting? Well, they generalize symmetries and quasiperiodic tilings, and all that.

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