Difference between revisions of "Boolean lattice"
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Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>). | Given any [[set]] <math>S</math>, the '''boolean lattice''' <math>B(S)</math> is a [[partially ordered set]] whose elements are those of <math>\mathcal{P}(S)</math>, the [[power set]] of <math>S</math>, ordered by inclusion (<math>\subset</math>). | ||
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| + | Every boolean lattice is a [[distributive lattice]], and the poset operations [[meet]] and [[join]] correspond to the set operations [[union]] and [[intersection]]. | ||
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| − | + | [[Category:Definition]] | |
Latest revision as of 21:46, 20 April 2008
Given any set 
, the boolean lattice 
is a partially ordered set whose elements are those of 
, the power set of , ordered by inclusion (
).
When has a finite number of elements (say 
), the boolean lattice associated with is usually denoted 
. Thus, the set 
is associated with the boolean lattice 
with elements 
and 
, among which 
is smaller than all others, is larger than all others, and the other six elements satisfy the relations 
, 
, 
and no others.
The Hasse diagram for is given below:
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Every boolean lattice is a distributive lattice, and the poset operations meet and join correspond to the set operations union and intersection.
This article is a stub. Help us out by expanding it.
