TITLE: Axioms for atomless Boolean algebras
QUESTION [6 upvotes]: I'm embarrassed to be asking, but:
"Write down a set of axioms for the theory of atomless Boolean algebras."
This is Exercise 1.14 in Chapter 9 of "Models and Ultraproducts" by Bell and Slomson. I'm trying to read this on my own. I have no math community except this one.
Clearly we need the axioms for Boolean algebras, and then at least one more. The additional one might say something like: 
For any non-zero element x, there's a non-zero element y such that 0 < y < x.
But I don't know if that's anywhere close to correct. My further problem is that even if it were, I wouldn't know how to prove I had an axiom system for atomless Boolean algebras. Thanks for any help.
"Clueless in Tucson"

REPLY [3 votes]: If we want to axiomatize Boolean algebras in terms of partial order, we want to specify that the order is dense in the usual sense.  So a possible additional axiom is
$$\forall x \forall y(x<y\implies \exists z(x \lt z \land z \lt y)).$$
But this can be derived from the weaker-seeming axiom
 $$\forall y(\lnot(y=0) \implies \exists z(0 \lt z \land z \lt y))$$
that you suggested.  Thus your axiomatization is perfectly correct and complete (pun).