complete distributivity
A lattice^{} $L$ is said to be completely distributive if it is a complete lattice^{} such that, given any sets $K\subseteq I\times J$ such that $K$ projects onto $I$, and any subset $\{{x}_{ij}\mid (i,j)\in K\}$ of $L$,
$$\underset{i\in I}{\bigwedge}(\underset{j\in K(i)}{\bigvee}{x}_{ij})=\underset{f\in A}{\bigvee}(\underset{i\in I}{\bigwedge}{x}_{if(i)}),$$  (1) 
where $K(i):=\{j\in J\mid (i,j)\in K\}$, and $A=\{f:I\to J\mid f(i)\in K(i)\text{for all}i\in I\}$.
By setting $I=J=\{1,2\}$ and $K=\{(1,1),(2,1),(2,2)\}$, then $K(1)=\{1\}$, $K(2)=\{1,2\}$, and $A$ consists of two functions $\{(1,1),(2,1)\}$ and $\{(1,1),(2,2)\}$. Then, the equation above reads:
$$({x}_{11})\wedge ({x}_{21}\vee {x}_{22})=({x}_{11}\wedge {x}_{21})\vee ({x}_{11}\wedge {x}_{22})$$ 
which is one of the distributive laws, so that complete distributivity implies distributivity.
More generally, setting $I=\{1,2\}$ and $J$ containing $1$ but otherwise arbitrary, and $K=\{(2,j)\mid j\in J\}\cup \{(1,1)\}$. Then $K(1)=\{1\}$, $K(2)=J$, and $A$ is the set of functions from $I$ to $J$ fixing $1$, and the equation (1) above now looks like
$${x}_{11}\wedge (\bigvee \{{x}_{2j}\mid j\in J\}=\bigvee \{{x}_{11}\wedge {x}_{2j}\mid j\in J\}$$ 
which shows that completely distributivity implies join infinite^{} distributivity (http://planetmath.org/JoinInfiniteDistributive).
Remarks.

1.
Dualizing the above equation results in the same lattice. In other words, a completely distributive lattice may be equivalently defined using the dual of Equation (1). As a result, a completely distributive lattice also satisfies MID, and hence is infinite distributive.

2.
However, a complete^{} distributive lattice^{} does not have to be completely distributive. Here’s an example: let $\mathbb{N}$ be the set of natural numbers with the usual ordering^{}, and ${\mathbb{N}}^{\prime}$ be an identical copy of $\mathbb{N}$ such that each natural number^{} $n$ corresponds to ${n}^{\prime}\in {\mathbb{N}}^{\prime}$. Then ${\mathbb{N}}^{\prime}$ has a natural ordering induced by the usual ordering on $\mathbb{N}$. Take the union $N$ of these two sets. Then $N$ becomes a lattice if we extend the meets and joins on $\mathbb{N}$ and ${\mathbb{N}}^{\prime}$ by additionally setting
$${a}^{\prime}\vee b:=\{\begin{array}{cc}{a}^{\prime}\hfill & \text{if}b\le a,\hfill \\ {b}^{\prime}\hfill & \text{otherwise.}\hfill \end{array}$$ and
$${a}^{\prime}\wedge b:=\{\begin{array}{cc}b\hfill & \text{if}b\le a,\hfill \\ a\hfill & \text{otherwise.}\hfill \end{array}$$ Finally, let $L$ be the lattice formed from $N$ by adjoining an extra element $\mathrm{\infty}$ to be its top element. It is not hard to see that $L$ is complete and distributive. However, $L$ is not completely distributive, for ${0}^{\prime}\wedge (\bigvee \{a\mid a\in \mathbb{N}\})={0}^{\prime}\wedge \mathrm{\infty}={0}^{\prime}$, whereas $\bigvee \{{0}^{\prime}\wedge a\mid a\in \mathbb{N}\}=\bigvee \{0\}=0\ne {0}^{\prime}$.

3.
In some literature, completeness assumption^{} is not required, so that the equation (1) above is conditionally defined. In other words, the equation is defined only when each side of the equation exists first.

4.
Another generalization^{} is the socalled $(\U0001d52a,\U0001d52b)$distributivity, where $\U0001d52a$ and $\U0001d52b$ are cardinal numbers^{}. Specifically, a lattice $L$ is $(\U0001d52a,\U0001d52b)$distributive if it is complete and equation (1) is true whenever $I$ has cardinality $\le \U0001d52a$ and each $K(i)$ has cardinality $\le \U0001d52b$ for each $i\in I$.
References
 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title  complete distributivity 

Canonical name  CompleteDistributivity 
Date of creation  20130322 15:41:34 
Last modified on  20130322 15:41:34 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  22 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06D10 
Related topic  JoinInfiniteDistributive 
Defines  completely distributive 
Defines  
Defines  n)distributive 