What are lattices used for?
See for instance this book: Lattice Theory with Applications, Vijay K. Garg, which starts off as follows:


  Partial order and lattice theory now play an important role in many disciplines of computer science
  and engineering. For example, they have applications in distributed computing (vector clocks,
  global predicate detection), concurrency theory (pomsets, occurrence nets), programming language
  semantics (fixed-point semantics), and data mining (concept analysis). They are also useful in other
  disciplines of mathematics such as combinatorics, number theory and group theory. In this book,
  I introduce important results in partial order theory along with their applications in computer
  science. The bias of the book is on computational aspects of lattice theory (algorithms) and on applications (esp. distributed systems).


The book doesn't seem to mention recursion theory (theory of computable sets), but from Wikipedia's article on Computability theory, we see:


  When Post defined the notion of a simple set as an r.e. set with an infinite complement not containing any infinite r.e. set, he started to study the structure of the recursively enumerable sets under inclusion. This lattice became a well-studied structure. Recursive sets can be defined in this structure by the basic result that a set is recursive if and only if the set and its complement are both recursively enumerable. Infinite r.e. sets have always infinite recursive subsets; but on the other hand, simple sets exist but do not have a coinfinite recursive superset. Post (1944) introduced already hypersimple and hyperhypersimple sets; later maximal sets were constructed which are r.e. sets such that every r.e. superset is either a finite variant of the given maximal set or is co-finite. Post's original motivation in the study of this lattice was to find a structural notion such that every set which satisfies this property is neither in the Turing degree of the recursive sets nor in the Turing degree of the halting problem. Post did not find such a property and the solution to his problem applied priority methods instead; Harrington and Soare (1991) found eventually such a property.


Further reading, see the blog post Lattice Theory for Programmers and Non Computer Scientists.