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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?

Squirtle
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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.

Kevin Carlson
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  • That’s it! I forgot that obvious step! Therefore the only options are the various combinations of the atoms. Thank you very much – Squirtle Jul 28 '19 at 02:04