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 where "F" is the set of choice functions "f" choosing for each index "j" of "J" some index "f"("j") in "K". Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. 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" equal the top element of "L" for all indices "j" and "k" with all of the sets "K" 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": 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 formula_3 such  that for every completely distributive lattice "M" and monotonic function formula_4, there is a unique complete homomorphism formula_5 satisfying formula_6. For every poset "C", the free completely distributive lattice over a poset "C" exists and is unique up to isomorphism. 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".