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, $a\le b$ 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 $\bigvee \{2\wedge p\mid p\in M\}=\bigvee \{1\}=1\ne 2$.
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 (http://planetmath.org/CompleteDistributivity). 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

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It can be shown that any complete Boolean lattice is infinite distributive.
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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.
Title  joininfinite distributive 
Canonical name  JoininfiniteDistributive 
Date of creation  20130322 19:13:48 
Last modified on  20130322 19:13:48 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06D99 
Synonym  JID 
Synonym  MID 
Related topic  MeetContinuous 
Related topic  CompleteDistributivity 
Defines  meetinfinite distributive 
Defines  joininfinite identity 
Defines  meetinfinite identity 
Defines  infinite distributive 
Defines  countably distributive 
Defines  joincountable distributive 
Defines  meetcountable distributive 