Some more Problems about Orderings of Ultrafilters

TL;DR

This paper explores the relationships between different orderings of ultrafilters and their implications for compactness in logic and topology, introducing new orders and characterizations.

Contribution

It provides a model-theoretic characterization of the Comfort order and introduces a new order inspired by abstract model theory.

Findings

If E is (λ, λ)-regular and D is not, then E is not less than or equal to D.

Connections between ultrafilter orders and compactness properties are established.

Many open problems related to ultrafilter orderings are presented.

Abstract

We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We introduce a new order motivated by considerations in abstract model theory. For each of the above orders, we show that if $E$ is a $(\lambda, \lambda)$-regular ultrafilter, and $D$ is not $(\lambda, \lambda)$-regular, then $E \not \leq D$. Many problems are stated.