Does there exist a boolean lattice without atoms?

Question

There are atomic boolean lattices and this is the same as atomistic boolean lattices.

Does there exist a boolean lattice without atoms? (except of the degenerative case of one-element lattice)

Or at least give me an example of a non-atomic boolean lattice.

Answer 1

Take the (atomic) Boolean algebra of all subsets of the set $\mathbb N$ of natural numbers, and form a quotient algebra by identifying any two elements $a$ and $b$ whose symmetric difference $a\triangle b$ is finite. The result is a Boolean algebra without atoms.