## You are here

Homeexample of Boolean algebras

## Primary tabs

# 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. 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. 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)

## Mathematics Subject Classification

06B20*no label found*03G05

*no label found*06E05

*no label found*03G10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections