TITLE: Uniquely complemented but not Boolean
QUESTION [6 upvotes]: What is an example of a lattice $(L,\leq)$ that is uniquely complemented, but not Boolean?

REPLY [9 votes]: As mentioned in the comments, there are and this follows from Dilworth's construction of free uniquely complemented lattices. A simpler construction was later given by Chen and Grätzer [On the construction of complemented lattices, J. Algebra 11 (1969), 56–63]. There is a nice overview by George Grätzer in the AMS Notices in 2007: "Two Problems That Shaped a Century of Lattice Theory". Here is an excerpt that indicates that there were no other examples of non-distributive uniquely complemented lattices in 2007; I don't know if any new examples have been found in the last decade.

All known examples of nondistributive uniquely
  complemented lattices are freely generated,
  one way or another. Is there a construction of
  a nondistributive uniquely complemented lattice
  that is different?
In the same vein, is there a “natural” example of
  a nondistributive uniquely complemented lattice
  from geometry, topology, or whatever else?