Equation (1) is called the join-infinite identity, or JID for short. We also call a JID lattice.
If 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 of non-negative integers ordered by division, that is, iff . Then is a distributive lattice. However, fails JID, for if is the set of all odd primes, then , so , where as .
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 is said to be meet-infinite distributive if it is complete, and for any element and any subset of , we have
Equation (2) is called the meet-infinite identity, or MID for short. 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.
It can be shown that any complete Boolean lattice is infinite distributive.
|Date of creation||2013-03-22 19:13:48|
|Last modified on||2013-03-22 19:13:48|
|Last modified by||CWoo (3771)|