A family of elements of α is directed (with respect to a relation ≼ on α)
if there is a member of the family ≼-above any pair in the family.
A subset of α is directed if there is an element of the set ≼-above any
pair of elements in the set.
A monotone function on a sup-semilattice is directed.
An antimonotone function on an inf-semilattice is directed.
A preorder is a directed_order if for any two elements i, j
there is an element k such that i ≤ k and j ≤ k.
i ≤ k
j ≤ k