join-infinite distributive

A lattice $L$ is said to be join-infinite 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 join-infinite identity, or JID for short. We also call $L$ a JID lattice.

If $M$ is any two-element 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 non-negative 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 meet-infinite 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 meet-infinite 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 join-infinite and meet-infinite 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 infinite-distributivity is that of countable-distributivity: a lattice is join-countable distributive if JID holds for all countable subsets $M$ of $L$, and meet-countable 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.

 Title join-infinite distributive Canonical name JoininfiniteDistributive Date of creation 2013-03-22 19:13:48 Last modified on 2013-03-22 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 meet-infinite distributive Defines join-infinite identity Defines meet-infinite identity Defines infinite distributive Defines countably distributive Defines join-countable distributive Defines meet-countable distributive