I would like to see an example of a complete lattice $C$ which is both a frame and a dual-frame, i.e. finite meets distribute over arbitrary joins and finite joins distribute over arbitrary meets (<http://en.wikipedia.org/wiki/Complete_Heyting_algebra>), but $C$ is not completely distributive (<http://en.wikipedia.org/wiki/Completely_distributive_lattice>), i.e. arbitrary meets do not distribute over arbitrary joins (complete distributivity is a self-dual property). 


I was unable to find this example in Gierz et al. "Compendium of continuous lattices" or in other standard references. 



Counterexample on completely distributive lattices
A complete Boolean algebra is both a frame and a co-frame. It is completely distributive if and only if it is atomic (see for example [here](http://thue.stanford.edu/bool.html)), so all you need is a non-atomic complete Boolean algebra. The lattice of regular open sets (an open set is *regular* if it is equal to the interior of its closure) in say $[0, 1]$ is an example; this is complete because meets are calculated just as they are in the frame of all open sets.


