Completely distributive lattice

A complete lattice in which the identity

$$ \inf _ {i \in I } \ \sup _ {j \in J _ {i} } \ a _ {i,j} = \ \sup _ {f \in F } \ \inf _ {i \in I } \ a _ {i, f ( i) } $$

(called the complete distributive law) holds for all doubly-indexed families of elements $ \{ {a _ {i,j} } : {i \in I, j \in J } \} $, where $ F $ is the set of all choice functions for the family of sets $ \{ {J _ {i} } : {i \in I } \} $. Like the finite distributive law (see Distributive lattice), the complete distributive law is equivalent to its dual; that is, a lattice $ A $ is completely distributive if and only if the opposite lattice $ A ^ { \mathop{\rm op} } $ is completely distributive. Completely distributive lattices may also be characterized as those complete lattices in which every element $ a $ is expressible as the supremum of elements $ b $ such that, whenever $ S $ is any subset of $ A $ with $ \sup S \geq a $, there exists an $ s \in S $ with $ s \geq b $[a1]. Any complete totally ordered set is completely distributive; a complete Boolean algebra is completely distributive if and only if it is isomorphic to the full power set of some set. In general, a complete lattice is completely distributive if and only if it is imbeddable in a full power set by a mapping preserving arbitrary sups and infs.

References

[a1]	G.N. Raney, "Completely distributive complete lattices" Proc. Amer. Math. Soc. , 3 (1952) pp. 677–680
[a2]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)