Is this a finite Boolean algebra?

Question

Suppose in a Boolean algebra that the finite sum of all the atoms gives $1$. Does it follow that this Boolean algebra is finite?

I was motivated by the following: Does there exist an infnite Boolean algebra with only one atom?

Answer 1

Yes. Given such a Boolean algebra $B$ with finitely many atoms $a_i$ such that $\vee_i a_i=1$, if $b\ B$ then we have $$b=b\wedge \vee a_i=\vee b\wedge a_i.$$ Each term in the latter join is either $0$ or an atom, so $b$ is a finite join of atoms, and a simple cardinality argument shows $B$ must be finite.