# Completely distributive lattice

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