Axioms for atomless Boolean algebras

Question

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"

Answer 1

An atom is a minimal nonzero element. Your axiom says no such minimal element exists. Any model that satisfies the axioms is clearly a Boolean algebra. Could a model that satisfies your axiom have a minimal nonzero element? I would try a proof by contradiction.

Answer 2

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).