Completely distributive lattice

Properties

In addition, it is known that the following statements are equivalent for any complete lattice L:

• L is completely distributive.
• L can be embedded into a direct product of chains [0,1] by an order embedding that preserves arbitrary meets and joins.
• Both L and its dual order L^op are distributive continuous posets.

Direct products of [0,1], i.e. sets of all functions from some set X to [0,1] ordered pointwise, are also called cubes.

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 ${\displaystyle \varphi :C\to L}$ such that for every completely distributive lattice M and monotonic function ${\displaystyle f:C\to M}$, there is a unique complete homomorphism ${\displaystyle f_{\varphi }^{\ast }:L\to M}$ satisfying ${\displaystyle f=f_{\varphi }^{\ast }\circ \varphi }$. 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.

Examples

• The unit interval [0,1], ordered in the natural way, is a completely distributive lattice.
  • More generally, any complete chain is a completely distributive lattice.
• The power set lattice ${\displaystyle (\mathcal{P}(X),\subseteq )}$ for any set X is a completely distributive lattice.
• For every poset C, there is a free completely distributive lattice over C. See the section on Free completely distributive lattices above.