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>). | ||
| − | When <math>S</math> has a finite number of elements (say <math>|S| = n</math>), the boolean lattice associated with <math>S</math> is usually denoted <math>B_n</math>. Thus, the set <math>S = \{1, 2, 3\}</math> is associated with the boolean lattice <math>B_3</math> with elements <math>\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger than all others, and the other six elements satisfy the relations <math>\{1\}, \{2\} \subset \{1,2\}</math>, <math>\{1\}, \{3\} \subset \{1,3\}</math>, <math>\{2\}, \{3\} \subset \{2,3\}</math> and no others. | + | When <math>S</math> has a [[finite]] number of elements (say <math>|S| = n</math>), the boolean lattice associated with <math>S</math> is usually denoted <math>B_n</math>. Thus, the set <math>S = \{1, 2, 3\}</math> is associated with the boolean lattice <math>B_3</math> with elements <math>\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}</math> and <math>\{1, 2, 3\}</math>, among which <math>\emptyset</math> is smaller than all others, <math>S = \{1, 2, 3\}</math> is larger than all others, and the other six elements satisfy the relations <math>\{1\}, \{2\} \subset \{1,2\}</math>, <math>\{1\}, \{3\} \subset \{1,3\}</math>, <math>\{2\}, \{3\} \subset \{2,3\}</math> and no others. |
The [[Hasse diagram]] for <math>B_3</math> is given below: | The [[Hasse diagram]] for <math>B_3</math> is given below: | ||
Revision as of 13:42, 24 October 2006
This article is a stub. Help us out by expanding it.
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.
