CONGRUENCE OF ULTRAFILTERS

Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are n...

Linear Ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are ‘‘continuous’’ with respect to U modulo the ideal of those that are ‘‘trivial’’ with respect to U forms a division ring E(U ). (These division rings can also be described as the endomorphism rings of the simple left End( X )-modules.) We study this and the dual construction, ...

Linear Analogues of Ultrafilters

Tukey Types of Ultrafilters

We investigate the structure of the Tukey types of ultrafilters on countable sets partially ordered by reverse inclusion. A canonization of cofinal maps from a p-point into another ultrafilter is obtained. This is used in particular to study the Tukey types of p-points and selective ultrafilters. Results fall into three main categories: comparison to a basis element for selective ultrafilters, ...

Neighborhoods, Ultrafilters, and Canonicity

This paper will remind the reader of neighborhood semantics for modal logics, compare them with relational semantics, and then look at some questions about neighborhood semantics that have been answered, and some which are still open. We will then introduce “ultrafilter semantics,” a way of expressing all sets over a canonical frame in an ‘effable’ way. This provides us with a conceptually easy...