There is a decent introduction in Peter Johnstone's "Stone Spaces"
(Cambridge University Press), written from the point of view that
completely distributive lattices are (up to isomorphism) exactly the
Scott topologies on continuous posets (a.k.a. continuous dcpos, i.e.
continuous directed complete partial orders).

Johnstone uses the standard definition. However, there is a problem with
it in that you end up using the axiom of choice a lot.

There is also a good definition of "constructively completely
distributive lattice", classically equivalent to the standard one, that
has been studied (first) by Fawcett and Wood and (subsequently) by
Rosebrugh and Wood in a series of papers all entitled "Constructive
complete distributivity ..."

Fawcett and Wood "Constructive complete distributivity I" Math. Proc.
Cam. Phil. Soc. 107 (1990) 81-90.
Rosebrugh and Wood "Constructive complete distributivity II" Math.
Proc. Cam. Phil. Soc. 110 (1991) 245-249.

Another reference: Raney, "A subdirect-union representation for
completely distributive complete lattices", Proc. Amer. Math. Soc. 4
(1953) 518-522, showed how each completely distributive lattice could be
got (up to isomorphism) as the lattice of rounded upper sets in a set
with a dense transitive relation.

I developed these in my own paper "Information systems for continuous
posets", Theoretical Computer Science 114 (1993) 201-229..

Steve Vickers.


Janis Buls wrote:

>Would you recommend me
>literature about
>completely distributive lattices?
>
>Thank you.
>
>J. Buls
>buls@...
>
>
>