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joininfinite distributive
A lattice $L$ is said to be joininfinite distributive if it is complete, and for any element $x\in L$ and any subset $M$ of $L$, we have
$x\wedge\bigvee M=\bigvee\{x\wedge y\mid y\in M\}.$  (1) 
Equation (1) is called the joininfinite identity, or JID for short. We also call $L$ a JID lattice.
If $M$ is any twoelement set, then we see that the equation above is just one of the distributive laws, and hence any JID lattice is distributive. The converse of this statement is false. For example, take the set $N$ of nonnegative integers ordered by division, that is, $a\leq b$ iff $a\mid b$. Then $N$ is a distributive lattice. However, $N$ fails JID, for if $M$ is the set of all odd primes, then $\bigvee M=0$, so $2\wedge(\bigvee M)=2$, where as $\bigvee\{2\wedge p\mid p\in M\}=\bigvee\{1\}=1\neq 2$.
Also any completely distributive lattice is JID. The converse of this is also false. For an example of a JID lattice that is not completely distributive, see the last paragraph below before the remarks.
Dually, a lattice $L$ is said to be meetinfinite distributive if it is complete, and for any element $x\in L$ and any subset $M$ of $L$, we have
$x\vee\bigwedge M=\bigwedge\{x\vee y\mid y\in M\}.$  (2) 
Equation (2) is called the meetinfinite identity, or MID for short. $L$ is also called a MID lattice.
Now, unlike the case with a distributive lattice, where one distributive law implies its dual, JID does not necessarily imply MID, and vice versa. An example of a lattice satisfying MID but not JID can be found here. The dual of this lattice then satisfies JID but not MID, and therefore is an example of a JID lattice that is not completely distributive. When a lattice is both joininfinite and meetinfinite distributive, it is said to be infinite distributive.
Remarks

It can be shown that any complete Boolean lattice is infinite distributive.

An intermediate concept between distributivity and infinitedistributivity is that of countabledistributivity: a lattice is joincountable distributive if JID holds for all countable subsets $M$ of $L$, and meetcountable distributive if MID holds for all countable $M\subseteq L$.

When the sets $M$ in JID are restricted to filtered sets, then the lattice $L$ is join continuous. When $M$ are directed sets in MID, then $L$ is meet continuous.
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