Completely Distributive Lattice

In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.

Formally, a complete lattice L is said to be completely distributive if, for any doubly indexed family {xj,k | j in J, k in Kj} of L, we have

\begin{align}\bigwedge_{j\in J}\bigvee_{k\in K_j} x_{j,k} = \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}\end{align}

where F is the set of choice functions f choosing for each index j of J some index f(j) in Kj.

Complete distributivity is a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices.

Read more about Completely Distributive Lattice:  Alternative Characterizations, Properties, Free Completely Distributive Lattices, Examples

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