example of Boolean algebras

Below is a list of examples of Boolean algebras. Note that the phrase “usual set-theoretic 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 $\varnothing$ 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.

(product of algebras) Let $A$ and $B$ be Boolean algebras. Then $A\times B$ is a Boolean algebra, where

$\displaystyle(a,b)\vee(c,d)$ $\displaystyle:=$ $\displaystyle(a\vee c,b\vee d),$ (1)

$\displaystyle(a,b)\wedge(c,d)$ $\displaystyle:=$ $\displaystyle(a\wedge c,b\wedge d),$ (2)

$\displaystyle(a,b)^{\prime}$ $\displaystyle:=$ $\displaystyle(a^{\prime},b^{\prime}).$ (3)
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 non-empty 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 set-theoretic 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 set-theoretic ones: $U\wedge V=U\cap V$ , $0=\varnothing$ and $1=X$ . However, the join $\wedge$ and the complementation ${}^{\prime}$ on $A$ are different. Instead, they are given by

$\displaystyle U^{\prime}$ $\displaystyle:=$ $\displaystyle X-\overline{U},$ (4)

$\displaystyle U\vee V$ $\displaystyle:=$ $\displaystyle(U\cup V)^{\prime\prime}.$ (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

$\displaystyle(a,b)\vee(c,d)$	$\displaystyle:=$	$\displaystyle(a\vee c,b\vee d),$	(1)
$\displaystyle(a,b)\wedge(c,d)$	$\displaystyle:=$	$\displaystyle(a\wedge c,b\wedge d),$	(2)
$\displaystyle(a,b)^{\prime}$	$\displaystyle:=$	$\displaystyle(a^{\prime},b^{\prime}).$	(3)

	$\displaystyle U^{\prime}$	$\displaystyle:=$	$\displaystyle X-\overline{U},$		(4)
	$\displaystyle U\vee V$	$\displaystyle:=$	$\displaystyle(U\cup V)^{\prime\prime}.$		(5)