Isomorphism Classes of Generating Sets

TL;DR

This work studies the isomorphism types of generating sets for ultrafilters by constructing nonlinear, finite-support iterations of Mathias forcing indexed by directed posets $\mathbb{D}$. It shows how to realize simple $P_{\mathbb{D}}$-point ultrafilters on $\omega$ with bases isomorphic to $\lambda_0\times\lambda_1$ and extends the method to achieve higher-dimensional products under a supercompact cardinal, yielding normal ultrafilters on $\kappa$ with prescribed Tukey-type. The results provide a framework to control Tukey and depth spectra and to separate cardinal characteristics such as $\mathfrak b_\kappa$, $\mathfrak d_\kappa$, and $\mathfrak u_\kappa$, including models with large $2^\kappa$. Overall, the paper advances the understanding of how the combinatorial structure of generating sets shapes ultrafilter cofinal-types, in both standard and large-cardinal contexts, with broad implications for ultrafilter theory and related cardinal characteristics.

Abstract

We prove that for any two regular cardinals $ω<λ_0<λ_1$ there is a ccc forcing extension where there is an ultrafilter $U$ on $ω$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong λ_0\timesλ_1$. We use similar ideas to construct an ultrafilter with a base $\mathcal{B}$ as above which is order isomorphic to any given two-dimensional, well-founded, countably directed order with no maximal element. Similarly, relative to a supercompact cardinal, it is consistent that $κ$ is supercompact, and for any regular cardinals $κ<λ_0<λ_1<...<λ_n$, there is a ${<}κ$-directed closed $κ^+$-cc forcing extension where there is a normal ultrafilter $U$ on $κ$ with a base $\mathcal{B}$ such that $(\mathcal{B},\supseteq^*)\cong λ_0\times...\timesλ_n$. 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 [4]. Our construction also provides new models where $\mathfrak{u}_κ<2^κ$.

Figures (1)

• Figure 1: Models and ultrafilters of the $\lambda_0\times\lambda_1$-based iteration

Theorems & Definitions (81)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1
• Theorem 2.2