Ultrafilter spaces on the semilattice of partitions

The Stone–Čech compactification of the natural numbers βω (or equivalently, the space of ultrafilters on the subsets of ω) is a well-studied space with interesting properties. Replacing the subsets of ω by partitions of ω in the construction of the ultrafilter space gives non-homeomorphic spaces of partition ultrafilters corresponding to βω. We develop a general framework for spaces of this type and show that the spaces of partition ultrafilters still have some of the nice properties of βω, even though none of them is homeomorphic to βω. Further, in a particular space, the minimal height of a tree π-base and P-points are investigated