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 such that for every completely distributive lattice M and monotonic function, there is a unique complete homomorphism satisfying . 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.
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