1. Examples

1. ${\mbox{\bf 2}}^n$ ($n \geq 1$), the most general finite example up to isomorphism. In particular, 2 is an example.

   
   
   

2. Pow$(X)$ for any set $X$.

   
   
   

3. Any Boolean subalgebra of a Boolean algebra.

   
   
   

4. For any infinite set $X$, the lattice Pow$_ {fin} (X)$ of all finite and cofinite subsets of $X$.

   
   
   

5. Any interval $[a,b]$ of a Boolean algebra, with relativized operations.

   
   
   

6. Any direct product of Boolean lattices or Boolean algebras.

   
   
   

7. For any $n \geq 0$, the free Boolean algebra FBA$(n)$, which is isomorphic to    2$^ {2 ^ n}$, with the exponent $2 ^ n$ being just an integer.

   (In contrast, the free distributive lattice is    2$^ {{ \mbox{\bf 2}} ^ n}$ (lattice exponent    2$^ n$) with 0 and 1 deleted.)

   
   
   

8. For any Boolean algebra $B$ and lattice ideal $I$, the lattice $B/I$ of equivalence classes, where $x \equiv y$ means $x + y \in I$. (Recall that a nonempty subset $I$ of a lattice $L$ is an ideal if $I$ is a downset closed under joins.)

   
   
   

9. For any infinite set $X$, the lattice Pow$(X) /$   F of all subsets of $X$ modulo finite subsets. This means the lattice of all equivalence classes of subsets of $X$, where two subsets are considered equivalent if their symmetric difference [defined below] is finite.

   
   
   

10. The lattice of measurable subsets of the reals modulo sets of measure 0.

    
    
    

11. The lattice of equivalence classes of a first-order language, where equivalence means logical equivalence and the operations are ``and'', ``or'', and ``not''.

12. Clopen$(X)$, where $X$ is a topological space.

    
    
    

13. Any Boolean ring with 1, made into a Boolean algebra as below.