example of Boolean algebras
Below is a list of examples of Boolean algebras. Note that the phrase “usual settheoretic operations^{}” refers to the operations of union $\cup $, intersection^{} $\cap $, and set complement^{} ${}^{\prime}$.

1.
Let $A$ be a set. The power set^{} $P(A)$ of $A$, or the collection^{} of all the subsets of $A$, together with the operations of union, intersection, and set complement, the empty set^{} $\mathrm{\varnothing}$ and $A$, is a Boolean algebra^{}. This is the canonical example of a Boolean algebra.

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 $F(A)\cup cF(A)$ is a Boolean algebra.

3.
More generally, any field of sets is a Boolean algebra. In particular, any sigma algebra $\sigma $ in a set is a Boolean algebra.
 4.

5.
More generally, if we have a collection of Boolean algebras ${A}_{i}$, indexed by a set $I$, then ${\prod}_{i\in I}{A}_{i}$ is a Boolean algebra, where the Boolean operations are defined componentwise.

6.
In particular, if $A$ is a Boolean algebra, then set of functions from some nonempty set $I$ to $A$ is also a Boolean algebra, since ${A}^{I}={\prod}_{i\in I}A$.

7.
(subalgebras^{}) Let $A$ be a Boolean algebra, any subset $B\subseteq A$ such that $0\in B$, ${a}^{\prime}\in B$ whenever $a\in B$, and $a\vee b\in B$ whenever $a,b\in B$ is a Boolean algebra. It is called a Boolean subalgebra of $A$. In particular, the homomorphic image^{} of a Boolean algebra homomorphism is a Boolean algebra.

8.
(quotient algebras^{}) Let $A$ be a Boolean algebra and $I$ a Boolean ideal in $A$. View $A$ as a Boolean ring^{} and $I$ an ideal in $A$. Then the quotient ring $A/I$ is Boolean, and hence a Boolean algebra.

9.
Let $A$ be a set, and ${R}_{n}(A)$ be the set of all $n$ary relations on $A$. Then ${R}_{n}(A)$ is a Boolean algebra under the usual settheoretic operations. The easiest way to see this is to realize that ${R}_{n}(A)=P({A}^{n})$, the powerset of the $n$fold power of $A$.

10.
The set of all clopen sets in a topological space^{} is a Boolean algebra.

11.
Let $X$ be a topological space and $A$ be the collection of all regularly open^{} sets in $X$. Then $A$ has a Boolean algebraic structure^{}. The meet and the constant operations follow the usual settheoretic ones: $U\wedge V=U\cap V$, $0=\mathrm{\varnothing}$ and $1=X$. However, the join $\wedge $ and the complementation ${}^{\prime}$ on $A$ are different. Instead, they are given by
${U}^{\prime}$ $:=$ $X\overline{U},$ (4) $U\vee V$ $:=$ ${(U\cup V)}^{\prime \prime}.$ (5)
Title  example of Boolean algebras 

Canonical name  ExampleOfBooleanAlgebras 
Date of creation  20130322 17:52:33 
Last modified on  20130322 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 