The strong Prikry property

TL;DR

The paper introduces the strong Prikry property, a combinatorial condition on posets that guarantees the existence of an ultrafilter leading to a specific Boolean ultrapower phenomenon, unifying several known forcing notions.

Contribution

It defines the strong Prikry property and demonstrates its presence in various forcing notions, linking combinatorial properties to ultrafilter and ultrapower constructions.

Findings

Strong Prikry property implies existence of an ultrafilter on the Boolean algebra.

Prikry, Magidor, and generalized Prikry forcing satisfy this property.

Unified perspective on ultrapower constructions in different forcing contexts.

Abstract

I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this property: Prikry forcing, Magidor forcing and generalized Prikry forcing.