Properties of Laver forcing associated with a co-ideal expressed via the Katetov order

TL;DR

This work characterizes when Laver-type forcing derived from a co-ideal, $\mathbb{L}(\mathcal{I}^+)$, adds Cohen or random reals via the Katětov order, showing Cohen-reals occur precisely when $\mathsf{nwd} \leq_K \mathcal{I}\upharpoonright X$ for some $X\in\mathcal{I}^+$, and that random reals are never added. It develops a sharp criterion for the Laver property using a family of ideals $\mathcal{L}_f$ and proves that the Laver property does not coincide with the non-addition of Cohen reals, even for ultrafilters, with counterexamples and extensions to Miller-Mathias-type variants. The paper also explores adding half-Cohen reals via co-ideals, showing that if $\mathsf{HC} \leq_K \mathcal{I}\upharpoonright X$ for some $X$, then both $\mathbb{L}(\mathcal{I}^+)$ and $\mathbb{PT}(\mathcal{I}^+)$ add half-Cohen reals, while noting limitations of this approach. In the 1-1 or constant-property section, it confirms the property for $\mathbb{L}((\mathsf{fin}\times\mathsf{fin})^+)$, aligning with similar results in the field. The final section connects several cardinal invariants related to ideals, establishing equalities and inequalities that tie $\mathsf{add}$, $\mathsf{cov}$, $\mathsf{non}$ across specific ideals and reinforcing the structural link between Katětov order and forcing constructions.

Abstract

We study variants of classical Laver forcing defined from co-ideals and analyze their combinatorial properties in terms of the Katětov order. In particular, we give a Katětov-theoretic characterization of when Laver forcing associated with a co-ideal adds Cohen reals, and we show that such forcings never add random reals. Improving a result of Błaszczyk and Shelah we prove that the addition of Cohen reals and the Laver property are not equivalent, even in the case of ultrafilters. As an application, we investigate the problem of adding half Cohen reals without adding Cohen reals via tree-like forcings arising from co-ideals. We obtain several partial results and structural obstructions. Finally, we resolve several open questions from the literature concerning the one-to-one or constant property and cardinal invariants associated with ideals.

Theorems & Definitions (70)

• Theorem 1.1: Farah and Zapletal 4more
• Theorem 1.2: Sabok and Zapletal sabok
• Theorem 1.3: Błaszczyk and Shelah blaszczyk
• Theorem 1.4: Fernández-Bretón fernandez
• Definition 1.5
• Proposition 1.6: Folklore
• Theorem 2.1
• proof
• Claim 2.2
• Theorem 2.3