Let $V$ be a countable set of propositional variables, and $L=\{-,\vee,\wedge\}$ be a formalised language of the propositional calculus.
Let $A$ be the free Lindenbaum-Tarski algebra of formulas on the countable set of propositional variables.
Is A atomic or atomless? This is not clear for me because on one hand, $V$ is countable, but on the other hand, $L$ only consists of finitary operations.
I'd say that if $\phi$ is some element, and $v$ is any variable not appearing in it (a formula only has finitely many), that $\phi \land v$ is strictly smaller than $\phi$. So no atoms.