Boolean lattice
In this entry, the notions of a Boolean lattice, a Boolean algebra, and a Boolean ring are defined, compared and contrasted.
Boolean Lattices
A Boolean lattice is a distributive lattice in which for each element there exists a complement such that
In other words, a Boolean lattice is the same as a complemented distributive lattice. A morphism between two Boolean lattices is just a lattice homomorphism (so that and may not be preserved).
Boolean Algebras
A Boolean algebra is a Boolean lattice such that and are considered as operators (unary and nullary respectively) on the algebraic system. In other words, a morphism (or a Boolean algebra homomorphism) between two Boolean algebras must preserve and . As a result, the category of Boolean algebras and the category of Boolean lattices are not the same (and the former is a subcategory of the latter).
Boolean Rings
A Boolean ring is an (associative) unital ring such that for any , . It is easy to see that
-
•
any Boolean ring has characteristic , for ,
-
•
and hence a commutative ring, for , so , and therefore .
Boolean rings (with identity, but allowing 0=1) are equivalent to Boolean lattices. To view a Boolean ring as a Boolean lattice, define
To view a Boolean lattice as a Boolean ring, define
The category of Boolean algebras is naturally equivalent to the category of Boolean rings.
References
- 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
- 2 R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
Title | Boolean lattice |
Canonical name | BooleanLattice |
Date of creation | 2013-03-22 12:27:20 |
Last modified on | 2013-03-22 12:27:20 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 19 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 06E05 |
Classification | msc 03G05 |
Classification | msc 06B20 |
Classification | msc 03G10 |
Classification | msc 06E20 |
Synonym | Boolean algebra |
Related topic | BooleanRing |