In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.
For example, the following is an equivalent law that avoids the use of choice functions[citation needed].
This version of complete distributivity only implies the original notion when admitting the Axiom of Choice.
In addition, it is known that the following statements are equivalent for any complete lattice L:[2] Direct products of [0,1], i.e. sets of all functions from some set X to [0,1] ordered pointwise, are also called cubes.
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
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.