March 16
Shehzad Ahmed,
Ohio University
When is pcf(A) well behaved?
Recall that, for a set $A$ of regular cardinals, we define $$\operatorname{pcf}(A) := \{\operatorname{cf}(\prod A/D): D \text{ is an ultrafilter on A}\}.$$ In the case when $A$ is an interval of regular cardinals satisfying $|A|<\min (A)$, we can say quite a bit about how $\operatorname{pcf}(A)$ behaves. For example, we know that $\operatorname{pcf}(\operatorname{pcf}(A))=\operatorname{pcf}(A)$, and that we can get transitive generators for all $\lambda \in \operatorname{pcf}(A)$. We might then as ourselves what we can say if we remove one or both assumptions of these assumptions on $A$. That is, are there other assumptions under which $\operatorname{pcf}(A)$ is well behaved?
Throughout this talk, I will survey some of the literature regarding this question, and discuss a number of important open questions.