Directed schemes of ideals and cardinal characteristics. I: The meager additive ideal

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>ℳ</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>I</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>𝕀</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for the ideal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℳ𝒜</mml:mi> </mml:math> of meager-additive sets of reals. This directed scheme does not only helps us to understand more the combinatorics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℳ𝒜</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:math> and other classical characteristics. Furthermore, we demonstrate the consistency of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">cov</mml:mo> <mml:mo>(</mml:mo> <mml:mi>𝒩𝒜</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mi>𝔠</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">cof</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ℳ𝒜</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&lt;</mml:mo> <mml:mo form="prefix">non</mml:mo> <mml:mo>(</mml:mo> <mml:mi>𝒮𝒩</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The first one answers a question raised by the authors in [14].