# “Gaps” in the Rudin-Keisler ordering

If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) \setminus \{p\}$ contains a minimal element.)

Let $\text{NPU}(\omega)$ be the set of non-principal ultrafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by $${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$

It is easy to see that $\leq_{RK}$ is reflexive and transitive, but not anti-symmetric. Set ${\cal U}\simeq_{RK} {\cal V}$ if ${\cal U}\leq_{RK}{\cal V}$ and ${\cal V}\leq_{RK}{\cal U}$. So $\text{NPU}(\omega)/\simeq_{RK}$ is a poset with the Rudin-Keisler order applied to equivalence classes.

Does $\text{NPU}(\omega)/\simeq_{RK}$ contain elements that are the lower part of a gap?

Yes. Take U, V non-isomorphic Ramsey ultrafilters. Then there is no W that is RK-strictly in between $U$ and $U\cdot V$. The reason is the same as that given in Infima in the Rudin-Keisler ordering