complete distributivity
A lattice is said to be completely distributive if it is a complete lattice such that, given any sets such that projects onto , and any subset of ,
(1) |
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).
Remarks.
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1.
Dualizing the above equation results in the same lattice. In other words, a completely distributive lattice may be equivalently defined using the dual of Equation (1). As a result, a completely distributive lattice also satisfies MID, and hence is infinite distributive.
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2.
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
and
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 .
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3.
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.
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4.
Another generalization is the so-called -distributivity, where and are cardinal numbers. Specifically, a lattice is -distributive if it is complete and equation (1) is true whenever has cardinality and each has cardinality for each .
References
- 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).
Title | complete distributivity |
---|---|
Canonical name | CompleteDistributivity |
Date of creation | 2013-03-22 15:41:34 |
Last modified on | 2013-03-22 15:41:34 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 22 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 06D10 |
Related topic | JoinInfiniteDistributive |
Defines | completely distributive |
Defines | |
Defines | n)-distributive |