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 \phi:C\rightarrow L such that for every completely distributive lattice M and monotonic function f:C\rightarrow M, there is a unique complete homomorphism f^*_\phi:L\rightarrow M satisfying f=f^*_\phi\circ\phi.