The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC.

Axiom(NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one functions $f,g: \mathbb N \to \mathbb N$ such that $f(\mathcal D) = g(\mathcal E)$.

Where we define $f(\mathcal D) = \{A \subset \omega: f^{-1}(A) \in \mathcal D\}$. Equivalently $f(\mathcal D)$ is the unique ultrafilter generated by $\{f(D): D \in \mathcal D\}$

There is an equivalent version of the axiom.

Axiom(NCF II): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exists a finite-to-one monotone function $f :\mathbb N \to \mathbb N$ such that $f(\mathcal D) = f(\mathcal E)$.

One can ask if the same property holds over an upper-subset of $\mathbb N^*$: First define the preorder $\le$ on $\mathbb N^*$ where $\mathcal F \le \mathcal D$ means that $\mathcal F = f(\mathcal D)$ for some finite-to-one monotone function $f :\mathbb N \to \mathbb N$. Now fix some $\mathcal F \in \mathbb N^*$ and define $(\mathcal F \uparrow) = \{ \mathcal D \in \mathbb N^*: \mathcal F < \mathcal D \}$

Is anything known about the following proposition?

Axiom?:For any two free ultrafilters $\mathcal D, \mathcal E \in (\mathcal F \uparrow)$, there exists a finite-to-one monotone function $f :\mathbb N \to \mathbb N$ such that $f(\mathcal D) = f(\mathcal E) \in (\mathcal F \uparrow)$.

In particular:

Is the proposition consistent?

Does it follow from any well-known additional axioms?

Does it hold for any known $\mathcal F$ under ZFC?

Thanks in advance.

6more comments