where , and .
By setting and , then , , and consists of two functions and . Then, the equation above reads:
which is one of the distributive laws, so that complete distributivity implies distributivity.
More generally, setting and containing but otherwise arbitrary, and . Then , , and is the set of functions from to fixing , and the equation (1) above now looks like
which shows that completely distributivity implies join infinite distributivity (http://planetmath.org/JoinInfiniteDistributive).
However, a complete distributive lattice does not have to be completely distributive. Here’s an example: let be the set of natural numbers with the usual ordering, and be an identical copy of such that each natural number corresponds to . Then has a natural ordering induced by the usual ordering on . Take the union of these two sets. Then becomes a lattice if we extend the meets and joins on and by additionally setting
Finally, let be the lattice formed from by adjoining an extra element to be its top element. It is not hard to see that is complete and distributive. However, is not completely distributive, for , whereas .
In some literature, completeness assumption is not required, so that the equation (1) above is conditionally defined. In other words, the equation is defined only when each side of the equation exists first.
- 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
|Date of creation||2013-03-22 15:41:34|
|Last modified on||2013-03-22 15:41:34|
|Last modified by||CWoo (3771)|