Boolean lattice

In this entry, the notions of a Boolean lattice, a Boolean algebra, and a Boolean ringMathworldPlanetmath are defined, compared and contrasted.

Boolean Lattices

A Boolean lattice B is a distributive latticeMathworldPlanetmath in which for each elementMathworldMathworld xB there exists a complementPlanetmathPlanetmath xB such that

xx =0
xx =1
(x) =x
(xy) =xy
(xy) =xy

In other words, a Boolean lattice is the same as a complemented distributive lattice. A morphismMathworldPlanetmathPlanetmath between two Boolean lattices is just a lattice homomorphismMathworldPlanetmath (so that 0,1 and may not be preserved).

Boolean Algebras

A Boolean algebra is a Boolean lattice such that and 0 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 0,1 and . As a result, the category of Boolean algebras and the category of Boolean lattices are not the same (and the former is a subcategoryMathworldPlanetmath of the latter).

Boolean Rings

A Boolean ring is an (associative) unital ring R such that for any rR, r2=r. It is easy to see that

  • any Boolean ring has characteristicPlanetmathPlanetmath 2, for 2r=(2r)2=4r2=4r,

  • and hence a commutative ring, for a+b=(a+b)2=a2+ab+ba+b2=a+ab+ba+b, so 0=ab+ba, and therefore ab=ab+ab+ba=ba.

Boolean rings (with identityPlanetmathPlanetmathPlanetmathPlanetmath, but allowing 0=1) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to Boolean lattices. To view a Boolean ring as a Boolean lattice, define

xy=xy,xy=x+y+xy,and  x=1+x.

To view a Boolean lattice as a Boolean ring, define

xy=xy   and   x+y=(xy)(xy).

The category of Boolean algebras is naturally equivalent to the category of Boolean rings.


  • 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