boolean algebra with finite elements

Question

I need to define a boolean algebra with 8 elements. I know all the Axioms to define a binary boolean algebra but I don't know how to do that with 8 elements.

Someone can guide me please? Thanks.

Answer 1

There is essentially just one choice for a Boolean algebra with $2^n$ elements, that is, the power set of a three element set, say $\{1,2,3\}$, with respect to union (supremum), intersection (infimum) and complementation.

If you want another example, take the set of positive divisors of $30$ (any number of the form $pqr$ where $p$, $q$ and $r$ are distinct primes will do), with respect to lowest common multiple (supremum), greatest common divisor (infimum) and $d\mapsto 30/d$ as complementation.

However, these two examples give isomorphic Boolean algebras, because of the fact that any finite Boolean algebra $B$ is isomorphic to the power set of the set of $B_a$, the atoms of $B$: an atom is a minimal element in $B\setminus\{0\}$ with respect to the natural order relation on $B$.