# Blog Archives

# Topic Archive: Boolean algebras

# Boolean ultrapowers and the Bukovsky-Dehornoy phenomenon

I will present a criterion for when an ultrafilter on a Boolean algebra gives rise to the Bukovsky-Dehornoy phenomenon, namely that the intersection of all intermediate ultrapowers is equal to the the Boolean model. Time permitting, I will show that the Boolean algebras of Prikry and Magidor forcing satisfy the strong Prikry property, and that these forcings come with a canonical imitation iteration whose limit model is the Boolean ultrapower by a very canonical ultrafilter on their respective Boolean algebras.

# When is a Boolean ultrapower an ultrapower?

The Boolean ultrapower construction is a natural generalization of the classical ultrapower construction, but the Boolean ultrapower uses an ultrafilter on a complete Boolean algebra instead of a set. It was initially unknown as to whether in ZFC there exists a Boolean ultrapower which is not always isomorphic to a classical ultrapower. This problem was resolved in 1976 by Bernd and Sabine Koppelberg who constructed a Boolean ultrapower which is not an ultrapower. On the other hand, there does not seem to be any reference in the mathematical literature to atomless Boolean ultrapowers which are isomorphic to classical ultrapowers.

We shall first generalize the notion of a Boolean ultrapower to the notion of a BPA-ultrapower which is in a sense the most general ultrapower construction. Then by applying a result of Joel David Hamkins which characterizes the Boolean ultrapowers which are classical ultrapowers, we shall investigate examples of Boolean ultrapowers which are not classical ultrapowers as well as Boolean ultrapowers which are classical ultrapowers. For instance, I claim that under GCH every complete atomless Boolean algebra has an ultrafilter which gives rise to a Boolean ultrapower which is not a classical ultrapower. On the other hand, using the Keisler-Shelah isomorphism theorem, we may construct Boolean ultrapowers in ZFC on a fairly general class of Boolean algebras which are classical ultrapowers.

# Boolean algebras of elementary substructures

In his 1976 paper Haim Gaifman proved that for every set I, every model M of PA has an elementary end extension N such that Lt(N/M) is isomorphic to P(I). I will present a proof.

# Boolean subalgebras of the computable atomless Boolean algebra

An abstract of this talk will be added.