[Schémas dirigés d’idéaux et caractéristiques cardinales. I : L’idéal maigre-additif]
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}} = \lbrace \mathcal{M}_I : I\in \mathbb{I}\rbrace $ 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 $\operatorname{cov}(\mathcal{NA})<\mathfrak{c}$ and $\operatorname{cof}(\mathcal{MA})<\operatorname{non}(\mathcal{SN})$. The first one answers a question raised by the authors in [14].
Nous introduisons la notion de schéma orienté d’idéaux pour caractériser les idéaux particuliers sur les réels, qui provient d’une formalisation du cadre des idéaux de Yorioka pour les ensembles de mesure fortement nulle. Nous prouvons des théorèmes généraux pour les schémas orientés et proposons un schéma orienté $\vec{\mathcal{M}} = \lbrace \mathcal{M}_I : I\in \mathbb{I}\rbrace $ pour l’idéal $\mathcal{MA}$ des ensembles maigres-additifs de réels. Ce schéma orienté nous aide non seulement à mieux comprendre la combinatoire de $\mathcal{MA}$ et ses caractéristiques cardinales, mais nous fournit également de nouvelles caractérisations des nombres d’additivité et de cofinalité de l’idéal maigre des réels.
De plus, nous montrons les liens entre les caractéristiques associées à $\mathcal{M}_I$ et d’autres caractéristiques classiques. En outre, nous démontrons la cohérence de $\operatorname{cov}(\mathcal{NA})<\mathfrak{c}$ et $\operatorname{cof}(\mathcal{MA})<\operatorname{non}(\mathcal{SN})$. Le premier de ces deux résultats répond à une question soulevée par les auteurs dans [14].
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Keywords: Directed scheme of ideals, meager-additive ideal, cardinal characteristics, null-additive ideal, forcing models
Mots-clés : Schéma orienté d’idéaux, idéal maigre-additif, caractéristiques cardinales, idéal nul-additif, modèles de forcing
Miguel A. Cardona 1 , 2 ; Diego A. Mejía 3 ; Ismael E. Rivera-Madrid 1
CC-BY 4.0
@article{CRMATH_2026__364_G1_107_0,
author = {Miguel A. Cardona and Diego A. Mej{\'\i}a and Ismael E. Rivera-Madrid},
title = {Directed schemes of ideals and cardinal characteristics. {I:} {The} meager additive ideal},
journal = {Comptes Rendus. Math\'ematique},
pages = {107--125},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.807},
language = {en},
}
TY - JOUR AU - Miguel A. Cardona AU - Diego A. Mejía AU - Ismael E. Rivera-Madrid TI - Directed schemes of ideals and cardinal characteristics. I: The meager additive ideal JO - Comptes Rendus. Mathématique PY - 2026 SP - 107 EP - 125 VL - 364 PB - Académie des sciences, Paris DO - 10.5802/crmath.807 LA - en ID - CRMATH_2026__364_G1_107_0 ER -
%0 Journal Article %A Miguel A. Cardona %A Diego A. Mejía %A Ismael E. Rivera-Madrid %T Directed schemes of ideals and cardinal characteristics. I: The meager additive ideal %J Comptes Rendus. Mathématique %D 2026 %P 107-125 %V 364 %I Académie des sciences, Paris %R 10.5802/crmath.807 %G en %F CRMATH_2026__364_G1_107_0
Miguel A. Cardona; Diego A. Mejía; Ismael E. Rivera-Madrid. Directed schemes of ideals and cardinal characteristics. I: The meager additive ideal. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 107-125. doi: 10.5802/crmath.807
[1] Borel images of sets of reals, Real Anal. Exch., Volume 20 (1994/95) no. 2, pp. 536-558 | MR | DOI
[2] Covering games and the Banach–Mazur game: -tactics, Can. J. Math., Volume 45 (1993) no. 5, pp. 897-929 | DOI | MR | Zbl
[3] Closed measure zero sets, Ann. Pure Appl. Logic, Volume 58 (1992) no. 2, pp. 93-110 | DOI | MR | Zbl
[4] Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3 (Matthew Foreman; Akihiro Kanamori, eds.), Springer, 2010, pp. 395-489 | MR | DOI | Zbl
[5] Separating cardinal characteristics of the strong measure zero ideal, J. Math. Log. (2025) (Online ready) | DOI
[6] On cardinal characteristics associated with the strong measure zero ideal, Fundam. Math., Volume 257 (2022) no. 3, pp. 289-304 | DOI | MR | Zbl
[7] Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets, RIMS Kokyuroku, Volume 2315 (2025), pp. 1-24
[8] Continuum many different things: localisation, anti-localisation and Yorioka ideals, Ann. Pure Appl. Logic, Volume 175 (2024) no. 7, 103453, 58 pages | DOI | MR | Zbl
[9] Cardinal characteristics associated with small subsets of reals (2024) (To appear in Fundam. Math.) | arXiv | Zbl
[10] On cardinal characteristics of Yorioka ideals, Math. Log. Q., Volume 65 (2019) no. 2, pp. 170-199 | DOI | MR | Zbl
[11] Localization and anti-localization cardinals, RIMS Kokyuroku, Volume 2261 (2023), pp. 47-77
[12] More about the cofinality and the covering of the ideal of strong measure zero sets, Ann. Pure Appl. Logic, Volume 176 (2025) no. 4, 103537, 31 pages | DOI | MR | Zbl
[13] The covering number of the strong measure zero ideal can be above almost everything else, Arch. Math. Logic, Volume 61 (2022) no. 5–6, pp. 599-610 | DOI | MR | Zbl
[14] Uniformity numbers of the null-additive and meager-additive ideals, J. Symb. Log. (2025) (Online first) | DOI
[15] Strong measure zero sets, Notices Am. Math. Soc., Volume 26 (1979) no. 3, pp. A-280 (Abstract 79T-E25)
[16] Many different uniformity numbers of Yorioka ideals, Arch. Math. Logic, Volume 61 (2022) no. 5–6, pp. 653-683 | DOI | MR | Zbl
[17] Powers of transitive bases of measure and category, Proc. Am. Math. Soc., Volume 93 (1985) no. 4, pp. 719-729 | DOI | MR | Zbl
[18] Some operations related with translation, Colloq. Math., Volume 57 (1989) no. 2, pp. 203-219 | DOI | MR | Zbl
[19] Every null-additive set is meager-additive, Isr. J. Math., Volume 89 (1995) no. 1–3, pp. 357-376 | DOI | MR
[20] Compacts de fonctions mesurables et filtres non mesurables, Stud. Math., Volume 67 (1980) no. 1, pp. 13-43 | DOI | MR
[21] Generalized Galois–Tukey-connections between explicit relations on classical objects of real analysis, Set theory of the reals (Ramat Gan, 1991) (Haim Judah, ed.) (Israel Mathematical Conference Proceedings), Volume 6, Bar-Ilan University, 1993, pp. 619-643 | MR
[22] The cofinality of the strong measure zero ideal, J. Symb. Log., Volume 67 (2002) no. 4, pp. 1373-1384 | DOI | MR
[23] Strong measure zero and meager-additive sets through the prism of fractal measures, Commentat. Math. Univ. Carol., Volume 60 (2019) no. 1, pp. 131-155 | DOI | MR
[24] Meager-additive sets in topological groups, J. Symb. Log., Volume 87 (2022) no. 3, pp. 1046-1064 | DOI | MR
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