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

TL;DR

This paper introduces a new framework called directed schemes of ideals to analyze the meager-additive ideal on reals, leading to novel characterizations of its cardinal invariants and establishing consistency results related to classical set-theoretic characteristics.

Contribution

It develops the concept of directed schemes of ideals for the meager-additive ideal, providing new characterizations and connections to classical cardinal characteristics, and proves consistency results answering open questions.

Findings

New characterizations of additivity and cofinality of the meager ideal

Connections established between characteristics of $ ext{Meager-Additive}$ and classical invariants

Proved consistency of $ ext{cov}( ext{NA})< ext{c}$ and $ ext{cof}( ext{MA})< ext{non}( ext{SN})$

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.