Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters](https://en.wikipedia.org/wiki/Ultrafilter) 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.


It is [known](https://en.wikipedia.org/wiki/Ultrafilter#Ultrafilters_on_.E2.84.98.28.CF.89.29) that if ${\cal R}$ is a minimal element of [Ramsey ultrafilter](https://en.wikipedia.org/wiki/Ultrafilter#Ultrafilters_on_.E2.84.98.28.CF.89.29), then $[{\cal R}]\_{\simeq\_{RK}}$ is a minimal element of $\text{NPU}(\omega)/\simeq\_{RK}$.


**Question.** Suppose ${\cal U, V}\in \text{NPU}(\omega)$ are non-Ramsey , do $[{\cal U}]\_{\simeq\_{RK}}, [{\cal V}]\_{\simeq\_{RK}}$ have an infimum in $\text{NPU}(\omega)/\simeq\_{RK}$?



Infima in the Rudin-Keisler ordering
**This part only shows the existence of lower bounds, which is not the point. See edit.**
This is consistently true for example when the near coherence principle of ultrafilters holds. It says, for any two non-principal ultrafilters $U, V$ there exists a finite-to-one $f: \omega\to \omega$ such that $f(U)=f(V)$. Since $f$ is finite-to-one, it makes sure $f(U)=f(V)$ is not principal. 


**Edit:** the answer is consistently no. Suppose there exist two non-isomorphic Ramsey ultrafilters $U, V$, then we claim $U\cdot V$ and $V\cdot U$ do not have an infimum. This follows from the following facts


1. $U\cdot V, V\cdot U$ are neither P-point nor Q-point
2. $U\cdot V \simeq V\cdot U$ implies $U\simeq V$
3. The only elements that are potentially RK below $U\cdot V$ are $V\cdot U$, $U$, $V$ (or the principal ultrafilters)
4. So the common non-principal lower bounds for $U\cdot V, V\cdot U$ are $U, V$. But they are incomparable.


All of these can be found in Blass' thesis.


