# Completely distributive lattice

In the mathematical area of order theory, a **completely distributive lattice** is a complete lattice in which arbitrary joins distribute over arbitrary meets.

Formally, a complete lattice *L* is said to be **completely distributive** if, for any doubly indexed family
{*x*_{j,k} | *j* in *J*, *k* in *K*_{j}} of *L*, we have

where *F* is the set of choice functions *f* choosing for each index *j* of *J* some index *f*(*j*) in *K*_{j}.[1]

Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices.[1]

Without the axiom of choice, no complete lattice with more than one element can ever satisfy the above property, as one can just let *x*_{j,k} equal the top element of *L* for all indices *j* and *k* with all of the sets *K*_{j} being nonempty but having no choice function.

## Alternative characterizations

Various different characterizations exist. For example, the following is an equivalent law that avoids the use of choice functions. For any set *S* of sets, we define the set *S*^{#} to be the set of all subsets *X* of the complete lattice that have non-empty intersection with all members of *S*. We then can define complete distributivity via the statement

The operator ( )^{#} might be called the **crosscut operator**. This version of complete distributivity only implies the original notion when admitting the Axiom of Choice.

## Properties

In addition, it is known that the following statements are equivalent for any complete lattice *L*:[2]

*L*is completely distributive.*L*can be embedded into a direct product of chains [0,1] by an order embedding that preserves arbitrary meets and joins.- Both
*L*and its dual order*L*^{op}are continuous posets.

Direct products of [0,1], i.e. sets of all functions from some set *X* to [0,1] ordered pointwise, are also called *cubes*.

## Free completely distributive lattices

Every poset *C* can be completed in a completely distributive lattice.

A completely distributive lattice *L* is called the **free completely distributive lattice over a poset C** if and only if there is an order embedding
such that for every completely distributive lattice

*M*and monotonic function , there is a unique complete homomorphism satisfying . For every poset

*C*, the free completely distributive lattice over a poset

*C*exists and is unique up to isomorphism.[3]

This is an instance of the concept of free object. Since a set *X* can be considered as a poset with the discrete order, the above result guarantees the existence of the free completely distributive lattice over the set *X*.

## Examples

- The unit interval [0,1], ordered in the natural way, is a completely distributive lattice.[4]
- More generally, any complete chain is a completely distributive lattice.[5]

- The power set lattice
for any set
*X*is a completely distributive lattice.[1] - For every poset
*C*, there is a*free completely distributive lattice over C*.[3] See the section on Free completely distributive lattices above.

## References

- B. A. Davey and H. A. Priestey,
*Introduction to Lattices and Order*2nd Edition, Cambridge University Press, 2002, ISBN 0-521-78451-4 - G. N. Raney,
*A subdirect-union representation for completely distributive complete lattices*, Proceedings of the American Mathematical Society, 4: 518 - 522, 1953. - Joseph M. Morris,
*Augmenting Types with Unbounded Demonic and Angelic Nondeterminacy*, Mathematics of Program Construction, LNCS 3125, 274-288, 2004 - G. N. Raney,
*Completely distributive complete lattices*, Proceedings of the American Mathematical Society, 3: 677 - 680, 1952. - Alan Hopenwasser,
*Complete Distributivity*, Proceedings of Symposia in Pure Mathematics, 51(1), 285 - 305, 1990.