# Minimal cardinality of a filter base of a non-principal uniform ultrafilters

Let $$\kappa$$ be an infinite cardinal. An ultrafilter $${\cal U}$$ on $$\kappa$$ is said to be uniform if $$|R|=\kappa$$ for all $$R\in{\cal U}$$. If $${\cal U}$$ is a non-principal ultrafilter on $$\kappa$$, denote by $$b({\cal U})$$ the minimal cardinality that a filter base for $${\cal U}$$ can have.

If $${\cal U, V}$$ are non-principal uniform ultrafilters on $$\kappa$$, do we necessarily have $$b({\cal U}) = b({\cal V})$$?

Thanks to Joseph van Name for making me aware of uniform ultrafilters and the fact that this question is only (potentially) interesting when restricted to these.

• You should restrict your question to the uniform ultrafilters to avoid trivial counterexamples. An ultrafilter $\mathcal{U}$ on a cardinal $\kappa$ is said to be uniform if $|R|=\kappa$ whenever $R\in\mathcal{U}$. Here is a related question that answers the case for $\omega$. math.stackexchange.com/questions/1883710/… – Joseph Van Name Mar 10 '19 at 15:01
• Thanks @JosephVanName for your comment! - I will restrict the question to uniform ultrafilters – Dominic van der Zypen Mar 10 '19 at 16:57

Your number $$b(U)$$ is usually called the "character" of the ultrafilter $$U$$.
In general, there may be uniform ultrafilters on the same set with different characters. For example, it is consistent with $$2^{\aleph_0}=\aleph_2$$ that some ultrafilters have character $$\aleph_1$$, others $$\aleph_2$$.
(Of course, if $$2^\kappa=\kappa^+$$, then all uniform ultrafilters have character $$\kappa^+$$.)