Finding number of Boolean algebras

Question

How many Boolean algebras are there with four elements $0,1,a,b$ ?

I don't know how to proceed with this.

Any ideas ?

Answer 1

Hints:

Try to fill out the multiplication and addition tables.

For addition, $a+1$ can't be zero because 1 is its own additive inverse. $a+1$ can't be 1 because that would imply $a=0$. The sum can't be $a$ because that would imply $0=1$. So there is only the possibility that $a+1=b$.

From this you an deduce what $a+b, b+1$ and $ab$ have to be.

Once you have this, compare it to $F_2\times F_2$ where $F_2$ is the field of two elements.