Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal

TL;DR

The paper develops the notion of directed schemes of ideals to study peculiar ideals on the real line, focusing on the meager-additive ideal MA. It introduces a directed scheme vec{M} of ideals M_I whose intersection recovers MA, and establishes a robust framework connecting these schemes to classical cardinal characteristics via Tukey connections and relational systems. It proves key structural results, such as add(M_I) = add(M) and cof(M_I) = cof(M), and derives upper bounds for cof(MA) in terms of cof(M) and the directed scheme invariants. The authors also demonstrate the consistency of cov(NA) < c and cof(MA) < non(SN) using ccc forcing constructions, thereby highlighting new interactions between MA and standard cardinal invariants and providing new tools for analyzing meager-additive and related ideals.

Abstract

We introduce the notion of directed scheme of ideals to characterize peculiar ideals on the reals, which comes from a formalization of the framework of Yorioka ideals for strong measure zero sets. We prove general theorems for directed schemes and propose a directed scheme $\vec{\mathcal{M}} = \{\mathcal{M}_I \colon I\in\mathbb{I}\}$ for the ideal $\mathcal{MA}$ of meager-additive sets of reals. This directed scheme does not only helps us to understand more the combinatorics of $\mathcal{MA}$ and its cardinal characteristics, but provides us new characterizations of the additivity and cofinality numbers of the meager ideal of the reals. In addition, we display connections between the characteristics associated with $\mathcal{M}_I$ and other classical characteristics. Furthermore, we demonstrate the consistency of $\mathrm{cov}(\mathcal{NA})<\mathfrak{c}$ and $\mathrm{cof}(\mathcal{MA})<\mathrm{non}(\mathcal{SN})$. The first one answers a question raised by the authors in arXiv:2401.15364.

Figures (2)

• Figure 1: Diagram of the cardinal characteristics associated with $\mathcal{I}$. An arrow $\mathfrak x\rightarrow\mathfrak y$ means that (provably in ZFC) $\mathfrak x\le\mathfrak y$.
• Figure 2: Cichoń's diagram including the cardinal characteristics associated with our ideals, and $\mathop{\mathrm{add}}\nolimits(\mathcal{E}) = \mathop{\mathrm{add}}\nolimits(\mathcal{M})$ and $\mathop{\mathrm{cof}}\nolimits(\mathcal{E}) = \mathop{\mathrm{cof}}\nolimits(\mathcal{M})$ due to Bartoszyński and Shelah BS1992. The inequality $\mathop{\mathrm{non}}\nolimits(\mathcal{M}_I)\leq\mathop{\mathrm{non}}\nolimits(\mathcal{E})$ and $\mathop{\mathrm{cov}}\nolimits(\mathcal{E})\leq\mathop{\mathrm{cov}}\nolimits(\mathcal{M}_I)$ holds whenever $\sum_{k<\omega}2^{-|I_k|}<\infty$.

Theorems & Definitions (68)

• Theorem 1.2: CM23
• Theorem 1
• Theorem 1.4: GaMS
• Definition 1.5
• Lemma 1.6: e.g. CMR2
• Theorem 1.7
• Theorem 1.8
• Theorem 2: \ref{['thm:b1']}
• Corollary 1.10
• Theorem 3