Help me get hyped about lattices 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.
 A: If combinatorics counts as an application for you, you can take a look at Chapter 3 of Stanley's
Enumerative Combinatorics. http://www-math.mit.edu/~rstan/ec/ec1/
Formal concept analysis is another example: http://en.wikipedia.org/wiki/Formal_concept_analysis .
A: 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.

A: 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.
