Isomorphism Classes of Generating Sets

Abstract

We introduce a new class of ultrafilters which generalizes the well-known class of simple $P$-point ultrafilters. We prove that for any well-founded $\sigma$-directed partial order $\mathbb{D}$ there is a mild forcing extension where there is an ultrafilter $U$ on $\omega$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. On a measurable cardinal we prove a similar result: relative to a supercompact cardinal, it is consistent that $\kappa$ is supercompact, and for a $\kappa^+$-directed well-founded poset $\mathbb{D}$, there is a ${<}\kappa$-directed closed $\kappa^+$-cc forcing extension where there is a \emph{normal} ultrafilter $U$ on $\kappa$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong \mathbb{D}$. These are optimal results in the class of $P$-points and realize every potential structure of a $P$-point. We apply our constructions to obtain ultrafilters with controlled Tukey-type, in particular, an ultrafilter with non-convex Tukey and depth spectra is presented, answering questions from \cite{Benhamou_2024}. Our construction also provides new models where $\mathfrak{u}_\kappa<2^\kappa$, answering questions from \cite{Benhamou_Goldberg2025}.