Reference for the fact: elements as union of atoms in a Atomic Boolean lattice

Question

I need a reference to a book with the following statement: "In a Atomic Boolean Lattice every element is the union of the atoms under lie it". Does not matter if it is presented as a exercise.

Answer 1

See :

• Steven Givant & Paul Halmos, Introduction to Boolean algebras (2009), page 118 :

Lemma 3 [Tarski]. The following conditions on a Boolean algebra are equivalent.

(1) The algebra is atomic.

(2) Every element is the supremum of the atoms it dominates.

(3) The unit is the supremum of the set of all atoms.