What is an example of a lattice $(L,\leq)$ that is uniquely complemented, but not Boolean?



Uniquely complemented but not Boolean
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](http://www.ams.org/mathscinet-getitem?mr=232715), 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](http://www.ams.org/notices/200706/)". 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?
> 
> 
> 


