Completely Distributive Lattice - Alternative Characterizations

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.