Does there exist an infnite Boolean algebra with only one atom?

Question

So I know that there exists an infinite Boolean algebra that has atoms, for example the power set on the natural numbers has an infinite number of atoms, ie the singleton sets.

But is there a Boolean algebra (infinite) which only has exactly one atom? Or would this imply some sort of contradiction? Or is there a trivial/canonical example of such?

Thanks

Answer 1

Let $B$ be any boolean algebra without atoms, such as the free boolean algebra on countably infinitely many generators. Let $B'$ be the algebra $2$. Then $B \times B'$ has precisely one atom, namely the element $(0,1)$.