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

Famous quotes containing the word completely:

    “After, when they disentwine
    You from me and yours from mine,
    Neither can be certain who
    Was that I whose mine was you.
    To the act again they go
    More completely not to know.”
    —Robert Graves (1895–1985)