The generic ultrafilter added by ${(\FIN \times \FIN)}^{+}$

TL;DR

This paper studies the properties of a specific generic ultrafilter added by a particular quotient, revealing its non-basic generation, intermediate Tukey type, and the nature of Tukey reductions to other ultrafilters.

Contribution

It characterizes the Tukey type of the generic ultrafilter added by  imes , showing it is not basically generated and has unique reduction properties.

Findings

Ultrafilter is not basically generated.

Ultrafilter does not have maximal Tukey type.

Tukey reductions are witnessed by Baire class one maps.

Abstract

We investigate the Tukey type of the generic ultrafilter added by the quotient $\mathcal{P}(\omega \times \omega) / (\mathrm{FIN} \times \mathrm{FIN})$. We prove that this ultrafilter is not basically generated and yet does not have the maximal Tukey type among directed partial orders of size continuum. Moreover, any Tukey reduction from this ultrafilter to any other ultrafilter is witnessed by a Baire class one map.