Algebra Seminar talk
2022-03-11
Jonathan Cancino
Some remarks about I-ultrafilters
Abstract:
Given an ideal I on $\omega$ (the set of natural numbers, or any countable set), an ultrafilter $U$ on $\omega$ is said to be an $I$-ultrafilter if for any function $f:\omega\to \omega$, there is $A\in U$ such that $f[A]\in I$.
This notion was introduced by James Baumgartner and it has proved to be very useful in the classification of combinatorial properties of ultrafilters, that is, many combinatorial properties of ultrafilters can be reformulated in terms of $I$-ultrafilters for a suitable ideal $I$. We will see some typical examples of this, and if time allows, we will review two cardinal invariants associated to the existence of such ultrafilters.