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.
Read more about this topic: Completely Distributive Lattice
Famous quotes containing the words free and/or completely:
“I am truly free only when all human beings, men and women, are equally free. The freedom of other men, far from negating or limiting my freedom, is, on the contrary, its necessary premise and confirmation.”
—Mikhail Bakunin (1814–1876)
“The bond between a man and his profession is similar to that which ties him to his country; it is just as complex, often ambivalent, and in general it is understood completely only when it is broken: by exile or emigration in the case of one’s country, by retirement in the case of a trade or profession.”
—Primo Levi (1919–1987)