Maximal Filters in the Lattice of Partitions of an Infinite Set

Abstract

We study maximal (proper) filters in the complete lattice of partitions of an infinite set~$X$. In the language of uniform spaces, these are precisely the atoms of the lattice of zero-dimensional uniformities on $X$, introduced by Pelant and Reiterman and studied further by Pelant, Reiterman, Rödl and Simon. The first half of this paper recovers, sharpens, and extends their classification in purely partition-theoretic terms. {Call a maximal filter of partitions {\em type I} if it does not contain all finite partitions, and {\em type II} if it does.} Type I filters are induced, in an essentially unique way, by ultrafilters on families of pairwise disjoint doubletons. {A type II filter} determines a non-principal ultrafilter on $X$, the \emph{heart}; the heart determines the filter precisely when {the former} is minimal in the Rudin--Keisler order. Each member $F$ of {a type II filter} gives rise to a closed \emph{fiber} in $X^{\ast }=\beta X\setminus X$, {consisting of} the set of ultrafilters agreeing with the heart on $F$. We prove a trichotomy describing the topology of arbitrary fibers. We then show that the fibers do not encode the filter: fibers do not form a semilattice under intersection, the closure of an infinite discrete set of ultrafilters of a single Rudin--Keisler type (a \emph{sparse} set, in our terminology) need not be a fiber, and a partition incompatible with a member of the filter may have a {fiber strictly larger than that member.} A representation that does succeed is nevertheless available in another category: the {type-II filters} with heart $\mathfrak{u}$ correspond to the maximal proper substructures of the ultrapower $X^X/\mathfrak{u}$ of the full structure on $X$. The topological representation problem remains open.

21 pages, LEAN 4 verification of all the new results