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

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

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

   ${(a,b)}^{\prime }$$:=$$(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: $U\wedge V=U\cap V$, $0=\mathrm{\varnothing }$ 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)