join-infinite distributive

A latticeMathworldPlanetmath L is said to be join-infinite distributive if it is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and for any element xL and any subset M of L, we have

xM={xyyM}. (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 converseMathworldPlanetmath of this statement is false. For example, take the set N of non-negative integers ordered by division, that is, ab iff ab. Then N is a distributive latticeMathworldPlanetmath. However, N fails JID, for if M is the set of all odd primes, then M=0, so 2(M)=2, where as {2ppM}={1}=12.

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 xL and any subset M of L, we have

xM={xyyM}. (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 ( 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.


  • 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 countableMathworldPlanetmath subsets M of L, and meet-countable distributive if MID holds for all countable ML.

  • 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