Rapid ultrafilters and summable ideals

TL;DR

This paper investigates the existence of ultrafilters that are not rapid but are associated with tall summable ideals, demonstrating their consistency under Martin's Axiom for -centered posets.

Contribution

It proves that assuming Martin's Axiom, ultrafilters exist for each tall summable ideal that are not rapid, addressing a specific open question.

Findings

Ultrafilters exist for each tall summable ideal under Martin's Axiom.

These ultrafilters are not necessarily rapid.

The result depends on set-theoretic assumptions.

Abstract

This note answers the following question: Is it consistent that for an arbitrary tall summable ideal I_g there exists an I_g-ultrafilter which is not rapid? We show that assuming Martin's Axiom for \sigma-centered posets such ultrafilters exist for each tall summable ideal I_g.