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.
Read more about this topic: Completely Distributive Lattice
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“If the alternative is to keep all just men in prison, or give up war and slavery, the State will not hesitate which to choose.”
—Henry David Thoreau (1817–1862)