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* = βX ∖ 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 u correspond to the maximal proper substructures of the ultrapower XX /u of the full structure on X. The topological representation problem remains open.
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BibTeX
@misc{feldman2026maximal,
title = {{Maximal Filters in the Lattice of Partitions of an Infinite Set}},
author = {David Victor Feldman and Alexander Wilce},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26201},
eprint = {2609.26201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26201},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal Filters in the Lattice of Partitions of an Infinite Set
AU - David Victor Feldman
AU - Alexander Wilce
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26201
UR - https://arxiv.org/abs/2609.26201
AB - 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^* = βX ∖ 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 u correspond to the maximal proper substructures of the ultrapower X^X /u of the full structure on X. The topological representation problem remains open.
ER -