example of Boolean algebras

Below is a list of examples of Boolean algebras. Note that the phrase “usual set-theoretic operationsMathworldPlanetmath” refers to the operations of union , intersectionMathworldPlanetmathPlanetmath , and set complementPlanetmathPlanetmath .

  1. 1.

    Let A be a set. The power setMathworldPlanetmath P(A) of A, or the collectionMathworldPlanetmath of all the subsets of A, together with the operations of union, intersection, and set complement, the empty setMathworldPlanetmath and A, is a Boolean algebraMathworldPlanetmath. This is the canonical example of a Boolean algebra.

  2. 2.

    In P(A), let F(A) be the collection of all finite subsets of A, and cF(A) the collection of all cofinite subsets of A. Then F(A)cF(A) is a Boolean algebra.

  3. 3.

    More generally, any field of sets is a Boolean algebra. In particular, any sigma algebra σ in a set is a Boolean algebra.

  4. 4.

    (productMathworldPlanetmathPlanetmathPlanetmathPlanetmath of algebrasMathworldPlanetmath) Let A and B be Boolean algebras. Then A×B is a Boolean algebra, where

    (a,b)(c,d) := (ac,bd), (1)
    (a,b)(c,d) := (ac,bd), (2)
    (a,b) := (a,b). (3)
  5. 5.

    More generally, if we have a collection of Boolean algebras Ai, indexed by a set I, then iIAi is a Boolean algebra, where the Boolean operations are defined componentwise.

  6. 6.

    In particular, if A is a Boolean algebra, then set of functions from some non-empty set I to A is also a Boolean algebra, since AI=iIA.

  7. 7.

    (subalgebrasPlanetmathPlanetmath) Let A be a Boolean algebra, any subset BA such that 0B, aB whenever aB, and abB whenever a,bB is a Boolean algebra. It is called a Boolean subalgebra of A. In particular, the homomorphic imagePlanetmathPlanetmathPlanetmath of a Boolean algebra homomorphism is a Boolean algebra.

  8. 8.

    (quotient algebrasPlanetmathPlanetmath) Let A be a Boolean algebra and I a Boolean ideal in A. View A as a Boolean ringMathworldPlanetmath and I an ideal in A. Then the quotient ring A/I is Boolean, and hence a Boolean algebra.

  9. 9.

    Let A be a set, and Rn(A) be the set of all n-ary relations on A. Then Rn(A) is a Boolean algebra under the usual set-theoretic operations. The easiest way to see this is to realize that Rn(A)=P(An), the powerset of the n-fold power of A.

  10. 10.

    The set of all clopen sets in a topological spaceMathworldPlanetmath is a Boolean algebra.

  11. 11.

    Let X be a topological space and A be the collection of all regularly openPlanetmathPlanetmath sets in X. Then A has a Boolean algebraic structurePlanetmathPlanetmath. The meet and the constant operations follow the usual set-theoretic ones: UV=UV, 0= and 1=X. However, the join and the complementation on A are different. Instead, they are given by

    U := X-U¯, (4)
    UV := (UV)′′. (5)
Title example of Boolean algebras
Canonical name ExampleOfBooleanAlgebras
Date of creation 2013-03-22 17:52:33
Last modified on 2013-03-22 17:52:33
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Example
Classification msc 06B20
Classification msc 03G05
Classification msc 06E05
Classification msc 03G10