Some more Problems about Orderings of Ultrafilters

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.